Heads on a Sunny Day: Newsletter Challenge (10/23/07)

Chances of it coming up heads on a sunny day is 50%, previous results are an average of a set of random results.

JD.

Just realised after reading post #2, If he flips the coin once a day then the average number of sunny days will also need to be considered, its not just the heads or tails sequence. Its more complicated than it first appears?

It sounds like it is sunny 70% of the time where he lives.

A normal coin will average out to coming up heads 50% of the time if your sample size is large enough.

So where's the challenge question?

Want to flip over it? Heads I win, tails you lose.

Since probability law/theory says that over a prolonged study, the coin will show 50% heads, 50% tails, the question is telling us that:

of the 50% of the throws where heads resulted 70% of these occured on sunny days (so 30% on not sunny days)

of the 50% of throws where tails resulted, 45% occured on sunny days (so 55% on not sunny days)

So....<big breath>

Head & sunny: 0.5 * 0.7 = 0.35
Tail & sunny: 0.5 * 0.45 = 0.225

From which I conclude there is a 0.075 probablity excess that Jack experiences a sunny day, i.e. the probability of a sunny day is 0.575. I'm off to roll a die to see if this is correct. Anyone got a pack of cards?

Jack should consider getting some sunglasses to ensure that he's not inadvertently biasing the results on a sunny day.

Ahem! But the challenge states that the coin is biased; so does not yield success (a head or non-head) in 1/2 ratio. Hence theory would state (not that any trend is valid or "by the book," no matter what, but) that Jack's trial results must be taken into consideration in order to establish a sample space for the conditional probablility that a head will come up, given that it is (already) a sunny day.

The (100 - 70 =) 30 (of 100) sunny days in which the coin did not come up heads during Jack's experiment represents 45% of some actual number of non-head tosses on sunny days. From this, a sample space size (population), along with success (failure) ratios for all possible events in that sample space, can be derived.

You can take it from there.

'Ahem! But the challenge states that the coin is biased '

No it doesn't...your understanding of the english language is flawed.

It states he 'suspects' it is biased on sunny days....not it 'is' biased.

"depends on what the definition of is, is". The only sure thing of it is, is, there are more sunny days than not. He probably doesn't live in Seattle.

that what is is that what is not is not is that it it is

add punctuation where you see fit......if you can

Making your assumption, it is still not possible to "take it from there" - the data can only correspond to different bias on sunny to that on not-sunny days, but there isn't appropriate data (e.g. proportion of tosses on different days) to determine the probability in either case.

Gentlemen:

Making no assumptions, really. The idea, had the initial encouragement to Rose been comprehended correctly, was that the "belief" ascribed to Jack, that the coin might be biased by sunniness (remember, that in probability all suppositions can be tested), is tantamount to an hypothesis to be tested--and that based on the data obtained by Jack's (again in probability wording) "experiment." It accomplishes nothing to presuppose any (so-called) "known" (or text book) probability (i.e., that heads and tails should come up equally based on countless trials not involving Jack's coin)--nor would it be reasonable (or in this instance possible) to continue Jack's experiment through countless trials in order to probabilistically prove or disprove that Jack's coin was biased depending on the weather conditions. Rather, Jacks data can be used to test the hypothesis--the belief and feeling leading to the statement (but not the assumption, because assumptions are not open to proof or disproof)--that the coin lands in such a way so as to suggest that it is biased. If it can be demonstrated that in Jack's experiment the frequency of "successes" (that a head will come up on a sunny day) exceeds the expected 1/2 success frequency (irrespective of cloudiness)by an amount not explainable (or test-able) as random variation, then there is reason to be confident, above mere belief, that the coin is indeed biased--and biased by the weather condition to boot.

Now, the challenge does not go that far; it only asks that we determine a probability, based on the available data, that a coin flip on a sunny day will land heads up--will it be greater than or less than the expected frequency of 1/2 for known unbiased coins? If one recognizes that the 45% of tosses coming up tails on sunny days corresponds to the 30% (the 100 - 70 percent) of not-heads (i.e., of tails) coming up on sunny days, then one has the means to determine a sample space, and from that a frequency of all possible (coin toss and weather) outcomes in the sample space. Then, from that point (i.e., knowing a sample space in which all possible events must occur), one can "take if from there" to calculate the singular conditional probability, P(H_eads|S_{unny days}), that any coin toss will come up heads given a toss on a sunny day: P(H ∩ S)/PS, or the probability of the event that "heads come up" AND "it is sunny" divided by the probability (among all possible events in the now known overall sample space) that it will be sunny. Again, the key is determining the sample space--without which no probability calculation is possible. Although my quick calculations might be a bit off--it was very late....) I came up with a sample space containing 67 events (not counting the events "neither heads nor tails" and "neither sunny nor cloudy"), including 21 successes (heads AND sunny) among 51 events (that it was sunny).

Finally, although Del's chivalry is noteworthy, it is worth bearing in mind that Rose might also have taken his (really unfounded) "attack" to mean that he held the opinion that she needed...defending--that she would not be able, without help, to (correctly) comprehend, or respond to, the reply to her post.

Who is this masked man who spouts such long winded diatribes...?

I'd love to respond but I'm afraid the urge to curl up and sleep overcame me before I reached the end.

But seriously...I'd challenge anyone to physically bias a standard shaped coin to produce those results. I'd wager it isn't possible... (1000 quatlous on the earth cat!)

Del

Please check that diatribe definition. It appears you are referring to yourself.

Oooooooooh have you had a humourectomy or something?

There is a serious point in each post... if you don't want to identify yourself or answer them then fine.

Am I bothered?

[Latin diatriba, learned discourse, from Greek diatribē, pastime, lecture.

Yup..learned discourse...that's me folks.

I shall post anonymously so you can't tell which of the great myriad of guests I am..

but I expect you have guessed the guest (that's a sort of joke thing...play on words..maybe look it up ?)

Learned,

Where, and in what discourse, did you learn your discourse exactly? Please, none of those fizzy answers.

Unwashed one.

Ok .

1. First I appologise for any offence.

2. Biased coin..I just put a substantial bend in a 2penny piece and flipped it 33 times ( I believe this is a reasonable sample size).

Results Heads 19 Tails 14 I think this is reasonable practical proof that a coin cannot be biased to give the results in the question whilst remaining a 'regular coin'.

Thus your assertion that the question states the coin is biased is wrong.

Or maybe I'm arguing with the wrong guest..that's why I would prefer it if you joined the forum where you would find us a much friendlier bunch.

Again I appologise for any offence. I only intended ammusement

Regards

Del

Just for the record. Coin details in case anyone else can be bothered to try...

Diameter ~ 26mm

Thicknes ~ 2mm

Bend across diameter to give a deviation of 2.5 mm at the edge.

That is a very obviously bent coin.

There are doubtless other ways to bias the coin but I challenge anyone to produce one which still looks like a coin and will give the results in the challenge question.

English pound coin shape with all weight on one edge, conical milling, and variable restitution coefficient. Any shape bouncy one side, extremely absorbent the other on a hard surface.

So that would still be considered a coin would it?.. it's still just theory...

I did the test got the results...

I'll donate £50 to charity if anyone can produce a 'coin' (still recognizable as such to a 'reasonable man in the street') which can give a 70% biased result in a normal flip onto a normal flat surface.

(For definition of normal...ask the aforementioned man)

I think that the question establishes that Jack is somewhat unusual - he defines days as either "sunny" or "not-sunny", he has a favourite coin, and rather than abandon it he starts on extended tests. Based on the way the results are presented, it is reasonable to assume he doesn't know a lot about statistics. However, given his apparently obsessive personality we should expect that he performs a very large number of experiments - more than sufficient to mean the 70% indicates that the probability really is significantly different from 50%.

So, what could cause such a large difference? I think we have to return to Jack's obsessive personality. He has to start from an identical configuration, and has developed a jig to flip the coin onto a rubber mat that his mother used for changing his nappies when he was a baby. Obviously, all coin flips have to be made between two and three in the afternoon. And he keeps this mat in the unshaded company greenhouse (I forgot to mention he worked in an organic foodstuff research company), that is used to investigate the response of plants to extreme conditions. So the average (small) number of bounces is very different on sunny and not-sunny days.

Oh, and Jack thinks he is normal...

No need for a special charitable donation, then - but don't forget the gift-aid form whenever you make one (mustn't deprive Gordon and Alastair of their opportunity to contribute).

P.S. I think it would also be possible to disguise a structure with very strange behaviour so that it at least looks like a coin - edges which are firm to the touch but collapse when bouncing, strange weight distribution - but you would certainly need to work hard at it.

P.S. I think it would also be possible to disguise a structure with very strange behaviour so that it at least looks like a coin - edges which are firm to the touch but collapse when bouncing, strange weight distribution - but you would certainly need to work hard at it.

It would be simple to make a coin with detachable faces so it could be either have two heads or two tails.

Even simpler still have 3 'favourite coins' one regular...one two headed and one two tailed.

PS.

I used to use the coin toss to tell me what I really wanted to do sometimes ..

The proceedure is heads you do A, tails you do B. (Now if you really want to do B but aren't yet aware of it...)

Toss the coin .. if it comes upheads (A) you find yourself saying 'make it best of 3' at which point you realise what you really want to do and off you go. I don't use the technique much today...but as an adolescent trying to pluck up courage to ask a girl on a date it came in handy! (TMI ?)

TMI?: Embarrass de Richesse, or just embarrass the cat (one of mine has just come and tried to indicate that should have read "embrace")

N.B. I usually stopped at the "do I really want to toss the coin" point. If I did, the options were equal and my built-in arbitrariness** did the trick.

**just ask my colleagues

Using a jig and starting from the same side!!

That defeats the whole purpose of randomality. The point of a coin toss is random results which means you toss not knowing the starting side of the coin and the amount of force the human hand delivers can very greatly, adding to the randomness.

Agreed. I said Jack must be weird. And the only other "remotely plausible" explanation suggested so far (that the difference was caused by Jack using a "heads" fall of the coin to decide if the day was sunny) has similar problems regarding Jack's understanding and use of randomness.

Ha maybe you've got it there......!

Like it...

"I can't decide if it's sunny or not!. I'll flip the coin...heads it's sunny"...self fullfilling prophecy...I toss more heads on a sunny day.

I bow to your lateralness and cunning... I rub around your shins in admiration.

Nice one...made me grin!

That must be it..I s'pose youll do some obscure maths on it now?

Brilliant!

Del

Unfortunately, I can't claim the credit - its' in someone else's posting somewhere, but I can't seem to find it now.

Unfortunately, I can't claim the credit - its' in someone else's posting somewhere, but I can't seem to find it now.

Well, let's see......way back in post number....SIX (6) !!!! I said something like:

On the other hand, perhaps the outcome of the coin flipping is influencing Jack's determination of the type of day. The challenge does not say whether Jack notes the type of day before or after the coin flip. Perhaps subconsciously he "wants" heads to be on "sunny" days. Therefore, it is Jack that is biased, and not the coin, as he unconsciously names more of the marginal days "sunny" after he sees that the coin has come up heads. Some would call this a "self-fulfilling prophecy".

Thanks - sorry for not acknowledging by name. There was also another (later) posting that postulated that when Jack could not decide whether the day was sunny or not, he used the flip of a coin to decide. If you make that assumption, plus a fair coin and statistically significant trials, it reduces the number of variables - so it becomes possible to can work out the proportion of intrinsic sunny and not sunny days, and also the proportion of initially uncertain days.

"so it becomes possible to can work out the proportion of intrinsic sunny and not sunny days"

Huh? Makes it possible or impossible? How can knowing that that data is subjective, biased, and flawed make it possible to "work out the proportion of intrinsic sunny and not sunny days"?

If he uses the coin to decide whenever he is not certain, and all other flips are after the weather has been decided, it becomes possible. His rationality has nothing to do with it - only the consistency** of his behaviour and the reduction in the number of variables to two (the probabilities of making flips to decide sunny or not, and the probabilities of flips on (once-decided) sunny days. (Dull day flips are of course 1-sunny).

N.B. So long as the only flaw is consistent bias, so long as you know what is that bias you can work back to the original data. That (for example) is what you do when you post-calibrate equipment.

**Obsessive behaviour may not be rational, but it is usually remarkably consistent.

I don't see anything other then 50/50, the other variable is the climate where he is located.

If he was standing in the Sahara he might have a sunny day everyday then he'd really be confused.

What a JERK!!!!!!!

The Physicist surely said:

"And the only other "remotely plausible" explanation suggested so far (that the difference was caused by Jack using a "heads" fall of the coin to decide if the day was sunny)"

You egocentric forum menace.

The fact that Jack's editor has restricted the scope of "day" for purposes of challenging us guys is only in order to make it possible for us to analyze (analyse) his situation; that is, the subset of events within the experimental sample space shall comprise two and only two subsets of events as pertains to the weather phenomena which Jack senses might be biasing the outcomes of his coin tosses--that would be the case whether there were many tosses or only enough tosses sufficient to yield the percentage results he reports. By saying that his days are defined in terms of whether or not it was sunny or not sunny (as he perceived on the day of any particular toss), tells us we need not be concerned with any other subset which could be taken into account with respect to what a day might be: nighttime; dawn or dusk; isolated clouds or rain; cloudy-bright; variably cloudy;.... Also, there is no basis in the challenge for assuming (there's that word again) a large number of trials by Jack; and consequently that the 70% ratio is in any way (apart from Jack's vague sense, which he asks us to help him with) a proof (or tending to proof) that his coin (or his weather ) might be bias-ing his outcomes. The 70% could describe a very large or relatively small number to trials. In sufficiency terms, there need have been only enough trials to arrive at 7 out of 10 flips (this prior to the knowledge that there were also results described, albeit indirectly, for un-sunny days).

Del, perhaps if I state it in a different way, you (both) might recognize that there was no reason to mince words over my implicit assertion, that Jack's "belief" about coin bias can be posited as a statement: that there is coin (or weather related coin) bias. The fact that Jack asks us to help him in regard to his "feeling" about the result, in itself, is just as if he has assumed that there is coin bias, but wants reassurance of that fact. It's as if he is challenging us to prove him right or wrong. Thus, for him to assert a belief that there is bias must be taken as the conditional statement--an hypothesis to be proved or disproved--that there indeed is bias. If, then, we can arrive at a probability based on his reported results that predicts with confidence that heads will come up more often than not on sunny days, we can tell Jack it will probably be more profitable to bet on head on a sunny day; if not...then not. The only remaining question then is, can we come up with a sample space (an overall set/count of discrete events which includes all possible coin-toss and weather-alternative outcomes) for his experiment based on the results he reported; in other words, is it possible to hypothecate (in the financial sense) a discrete number of possible toss event outcomes constituting all of the subsets of the sample space.

My sense was that there is enough information in the challenge to do this--which I explained above in the (what was not meant to be a) diatribe.

If time permits I will try to post a Venn diagram which illustrates my point. I say "try" because, as you no doubt are aware, diagrams can be tricky to post and still have them be legible; they are so sparse of content (vis-a-vis a photo) that the site's compression routines often eliminates too much necessary data.

henceforth: Lex (not an abbreviation for lectern or lectures...maybe lexicon though)

Yeah, what he said! (I think?)

It's a good thing these posts can only get so narrow or we'd really be in trouble at this point.

Reply to self and others:

I wll post at the bottom to give more room for image.

Thanks.

Lex

He's one of US!

Please declare yourselves and leave me out of this. (If you really think it's me, check the writing style, the excessive parade of standard formulae and the strangeness of numbers thereby derived)

Fyz

I agree with your analysis so far, however, you have not answered the original question.

The problem statement calls for the application of Bayes theorem, which deals with conditional proobabilities and states that:

P(H | sunny) = P(sunny | H) * P(H) / P(sunny)

where P (H | sunny) means "the probability of heads given it is sunny"

We are asked to determine P(H | sunny) and are given P(sunny | H) = 0.70 and P(sunny | T) = 0.45.

SIDE NOTE: Although these two probabilities add up to a total > 1.0, they are indeed possible (regardless of whether the coin is biased or not!); it all depends on the relative number of sunny vs. not sunny days.

It would be a simple matter to apply Bayes theorem as given above; however, we are not given quite enough information. One way to get around this limitation is to assume something about the fairness of the coin. In your case, you have assumed that the coin is fair (i.e., P(H) = P(T) = 0.5). Hence your result that 57.5% of days are sunny or P(sunny) = 0.575. [The general formula for finding the probability of sunny days given P(H) or P(T) is P(sunny) = P(H) * P(sunny | H) + P(T)*P(sunny | T).]

Going back to Bayes theorem allows us to answer the original question posed:

P(H | sunny) = P(sunny | H) * P(H) / P(sunny)

Assuming a fair coin, P(H) = 0.50 and P(sunny) = 0.575

therefore,

P(H | sunny) = 0.70 * 0.50 / 0.575 = 0.609

that is, on any given sunny day, the probability of flipping a head is 0.609 (60.9%) for a fair coin.

The problem can be solved for a biased coin in a similar fashion:

Step 1: Assume P(H), find P(T) = 1 - P(H)

Step 2: Find P(sunny) using formula given above

Step 3: Solve Bayes theorem equation given above

ER did, as you eventually recognise, answer the original question with her assumption of 50%. After that, you go on to calculate the probability of tossing a fair coin on a sunny day as 0.609. Hopefully, now you have had the opportunity to stand back form the algebra, you are kicking yourself (gently) because you can see that this result contradicts the assumption of a fair coin? And you'll get the same contradiction for any fixed probability and relative number of sunny-vs-not_sunny days you choose. That is because the numbers are simply not compatible with the distributions being the same on sunny and on not_sunny days - unless you either assume the results do not represent the statistics, or that Jack sometimes uses a flip of the coin to decide whether he calls the day sunny.

This result does NOT contradict the assumption of a fair coin! The conditional probability P (H | sunny) is not the same as P(H), the latter of which equals 0.5 for a fair coin. Also, you are incorrect that the numbers are "not compatible." Here's how it breaks down (I am using N = 1000 coin tosses = 1000 days):

500 H -> 350 on Sunny days (70%), 150 on Not Sunny days

500 T -> 225 on Sunny days (45%), 275 on Not Sunny days

To solve the problem, you need to "pivot" this chart around (I used Bayes' theorem in my previous analysis.):

350 + 225 = 575 Sunny days -> 350 H, 225 T

150 + 275 = 425 Not Sunny days -> 150 H, 275 T

Two things should now be clear:

1) There are more Sunny days (57.5%) than Not Sunny days.

2) P(H) = P(T) = 0.5 since he has flipped 500 H and 500 T.

3) P(H | Sunny) = # H on Sunny days/# Sunny dayss = 350/575 = 0.609 or 60.9% as I found perviously using Bayes' theorem. In other words, on a Sunny day during this 1000 day period, the probability that he flips a head is 0.609.

You appear to be saying that you will define a coin to be fair even though on sunny days it does not have a 0.5 probability of coming up heads. What else are you going to allow to affect this "fair" coin of yours? The choice of the sucker who is taking a bet with you as to whether the result will be heads or tails?

In case you still missed it - what I am saying is that either the flip of the coin is truly fair (0.5 probability of heads under any conditions), or it is not. In the former case, applying Bayes theorem results in a contradiction. In the latter case, neither in the question nor physical reality provide any basis for assuming that the probability of falling heads is 0.5 overall.
N.B. that the exact numbers in the question can be achieved with 10M+20N tosses (where M and N are arbitrary positive integers), corresponding to 7M+9N sunny days (and 3M+11N not-sunny). The situation you postulated corresponds to an additional constraint that M=2N.

As your teacher told you in scholl ( which I suspect you failed ) read the question, all of the question and you will have all of the information needed to answer the question.

What a waiste of time, arguing over shit that is not in the question, now we find the answer was soooo easy!!!!!!!!!!!!!!!!!!!!

I suggest you read the question carefully, and see whether you can find the "information" assumed in the "so easy" "answer".

Add to that: if we accept the information added in the "answer" that there were only 200 total flips used to derive these figures, a "good" answer does indeed become clear - but not the one provided.
The additional information provided simply tells us that the data was not adequately significant (in the statistical sense) even to say that the sunny-day probability was other than 50:50. (As approximately enumerated in #103, for example)

Hey, Fyz. I think you are correct, there is not enough information given. My original answer assumes, as many people did, and apparently, I say APPARENTLY (more on this next), so did the writer of the question, that the two groups of flips (heads and tails) were represented proportionally in the total by the 70% and 45%, since his answer numerically matches the answer I gave way back in post #6. I understand now that either one of the groups could have significantly more or less, and the percent "sunny" figures could still be valid.

Now, why did I say apparently? Well, we must assume that either the Challenge writer made the same mistake that I and others did in assuming the proportionality of flips, or perhaps something else was afoot. Knowing from past experience how the editors at CR4 work over the Challenge submissions, it is POSSIBLE, perhaps even PROBABLE, that the ORIGINAL draft of this question included the key piece of information needed to make the proportionality assumption valid, but got eliminated in editing down to fit the small space that CR4 requires.

KUDOS to you and the others for pointing out that there was not actually enough information to answer the question (or at least to come up with an answer matching the official one) with out assuming the proportionality of the flips.

I wonder if the writer of this challenge would care to respond on this issue?

You said:

P(H | sunny) = P(sunny | H) * P(H) / P(sunny)

Correct, but you said

Assuming a fair coin, P(H) = 0.50 and P(sunny) = 0.575

Incorrect. P(sunny) =0.7

So P(H | sunny) = P(sunny | H) * P(H) / P(sunny) = 0.7*0.5/0.7=0.5

No, the question gave proportion(sunnyIFheads)=0.7, and
it gave proportion(sunnyIFtails)=0.45.
***

Depending on the overall proportions of heads and tails, all that tells us is that the proportion of coin flips on sunny days lies between 0.45 and 0.7.
Unless the number of coin flips was adequately large, it tells us nothing at all about the probabilities.

However, I would agree on your ultimate numerical conclusion: if you make the quoted assumption that the coin is fair, that corresponds to a 0.5 probability that the coin will land heads up irrespective of other conditions.

*** I've steered clear of Baysian notation, partly because not everyone will know it, but also because I've observed that, although it has an attractive internal rigour, the Bayesian approach to statistics has caused all sorts of confusion; this would be acceptable if it could make any predictions that could not be found using a more naive sampling viewpoint. My observation in this thread is that, most [possibly all?] of the times that Bayes format has been used, the authors appear to have forgotten about the basics, such as what is presented information and what is assumption, and whether the results are statistically significant...

I've steered clear of Baysian notation, partly because not everyone will know it..

Yeah right. I probably wouldn't either, but some others here might know. Prove it. Doubtless you will never convince all Fyz (despite gallant efforts) but it seems worth asking. 'If a point is worth asking, it's worth asking well' as they say. Shame the original questioner didn't do so.

No, P(sunny) = 0.575; it's P(sunny | H) that = 0.7

What you are saying is that I made a major error but still came up with the correct solution!

Jack is blind. And his helpful neighbor is a liar.

It is probibly a psycholigical thing where he puts more energy into the flip on non-sunny days, which makes it flip just one more time to come up tails. Of course it could be that he is always starting from the same side all the time and he has excellent control over the flipping. This would skew the results more than he could know.

OK, so far no one has given this answer, so either I am smarter than the rest, or I am dumber than the rest! <grin>:

Statistically speaking, based on the data collected so far (could be a small sample) and since Jack is determining the sunny/not sunny conditions, the probability that Jack's coin will come up heads on a "sunny" day is about 61%.

How did I arrive a this (so far) unique answer? Well, if 70% of the time heads shows up it is "sunny", but 45% of the time that tails shows up it is "sunny", then if heads and tails are the only possible outcomes and all the sunny days are thus accounted for, and if it is "sunny" it will flip to either heads or tails, then heads will show up 70 / (70+45) = .60869565... or about 61% of the "sunny" days.

We THINK that there should be no relation between sunny days and head/tails flipping, but since flipping a single coin can always be either heads or tails, 50/50, there is some chance that a small sampling could be skewed by a random streak of "luck" falling within the sample. Increasing the sample size should reduce the error.

On the other hand, perhaps the outcome of the coin flipping is influencing Jack's determination of the type of day. The challenge does not say whether Jack notes the type of day before or after the coin flip. Perhaps subconsciously he "wants" heads to be on "sunny" days. Therefore, it is Jack that is biased, and not the coin, as he unconsciously names more of the marginal days "sunny" after he sees that the coin has come up heads. Some would call this a "self-fulfilling prophecy".

Another possibility is that there IS something physical about the coin that makes IT biased towards heads on a sunny day. For example, many coins are bi-metallic, i.e layers of two different alloys. Two different alloys may expand/contract with temperature changes at different rates, bending or otherwise distorting the flat shape of the coin. If the coin was left in a sunny spot where it warmed to a higher temperature on "sunny" days, the shape of the coin could be affected slightly, perhaps altering the normal 50/50 flipping outcomes. Although this is not likely, since most "clad" coins have the same alloy on both sides with a core of (nearly) pure copper, it could be a faulty blanks were used in minting the coins yielding a bias of the copper core towards one side or the other, with the "silvery" alloy being thick on one side and very thin on the other.

I would ask Jack to keep on flipping and see if his results were different after doubling, tripling, and quadrupling his original sample size. If there were no bias, I would bet that he would see a trend towards 50% of the time coming up heads on a sunny day, or on any day.

I like the explanation STL but there is one flaw in your reasoning.

"Another possibility is that there IS something physical about the coin that makes IT biased towards heads on a sunny day. For example, many coins are bi-metallic, i.e layers of two different alloys. Two different alloys may expand/contract with temperature changes at different rates, bending or otherwise distorting the flat shape of the coin. If the coin was left in a sunny spot where it warmed to a higher temperature on "sunny" days, the shape of the coin could be affected slightly, perhaps altering the normal 50/50 flipping outcomes. Although this is not likely, since most "clad" coins have the same alloy on both sides with a core of (nearly) pure copper, it could be a faulty blanks were used in minting the coins yielding a bias of the copper core towards one side or the other, with the "silvery" alloy being thick on one side and very thin on the other."

I am assuming that Jack is male. Most men carry their coins in their pocket. This would effectively warm the coin to body temperature and thus negate any bias on the flip unless the coin was left out to aclimate before the afore mentioned flip.

I am assuming that Jack is male. Most men carry their coins in their pocket.

I only said it was one possibility anyway. So you are presenting another possibility! In my scenario, Jack has set aside the particular coin for flipping, and so it lays on his desk, which happens to be near a window, sitting in sunshine on sunny days, but not on non-sunny days. I think this can be supported by the wording that calls the coin "his favorite coin". As such, it would not be mingled in with his other change in his pocket, but rest in a place of honor, or at least solitary confinement, in a prominent place, like on his desktop where he can reach it quickly to aid in his decision making!

OK i give........Uncle

Obviously, unless we take JD's reasonable assumption in post #1, we can't answer the question about the probability, as we know that Jack has not tested exhaustively (an infinite number of trials).

Moreover, even if we restrict the question further, to "what proportion of Jack's trials on sunny days came up heads", we still cannot answer the question. Let me give a couple of examples that show different proportions that fit the data:
a) Suppose that Jack tossed the coin a total of 300 times, and it came up heads 200 times and tails 100 times in total. Then he has 140 heads on sunny days, and 45 tails. Therefore, on a sunny day it came up heads 140 times out of 185, approximately 75.7% of the sunny-day trials.
b) Now suppose that it came up heads 100 times in all, and tails 200 times. Then, on sunny days it came up heads 70 times, and tails 90 times. That is heads 43.75% of trials on sunny days.
Both of these are possible - it just depends on how many of the trials were made on sunny days, how many on cloudy.

This is actually quite a good example of how you can mislead using statistical data; we must await the "answer" to see whether this was deliberate.

Fyz

I have to agree that this is a miss application of statistics. First the determination of sunny days can be subjective at best. And depending on where in the world you are you could have very few sunny days or inversely few non-sunny days, which would limit your samples on one side.

Also the possibility of that the total quatity of samples is too small is important.

Thirdly the question is vague and has missing information.

Ah ha..you could be onto something.

Say he always makes his decision on the basis ..

Heads gives me the 'fun/good/desired' result... Tails is the less desirable ouctcome..

Then his perception of wether the day is 'sunny' will be influenced by his enjoyable ativity.... e.g A 'marginal' day will be considered 'sunny' when playing golf but as 'non-sunny when clearing out loft or taking the mother in law shopping!

I claim immediate fame and fortune!

I claim immediate fame and fortune!

Hey Del, not so fast! I think I mentioned the possibility that Jack was biased and naming marginal days as "sunny" first! In my post #6 I said:

On the other hand, perhaps the outcome of the coin flipping is influencing Jack's determination of the type of day. The challenge does not say whether Jack notes the type of day before or after the coin flip. Perhaps subconsciously he "wants" heads to be on "sunny" days. Therefore, it is Jack that is biased, and not the coin, as he unconsciously names more of the marginal days "sunny" after he sees that the coin has come up heads. Some would call this a "self-fulfilling prophecy".

Share the wealth! You can keep the fame, just send me half of the fortune!

I'll go 60 / 40 As I hypothesised the heads is good theory

Well, don't forget those coins which land on their edge!

ROFL

Having a problem with the wording of the question, the coin comes up heads 70% of the time on sunny days, this infers that 30% of the time it came up tails not 45%? Then I can only reason that two separate columns are kept, one for sunny days and one for non sunny days? And if a coin can be expected to up heads 50% of the time then the 70% and 45% represents a relationship between the columns, or sunny days verses non sunny days? (45/70 * 100).So the next time he flips the coin it will be a 50/50 chance of heads but a 64.3% chance of it being sunny hypothetical.

JD.

I think the problem we have is believing the question, rather than the precision of the wording. Important aspects of the data have been omitted, presumably in an attempt to persuade us to draw false inferences. This disorients us, so we are apt to try to draw unsupportable conclusions from the data.

For this case: if we have a fair coin and sufficient trials, there should be equal number of heads as tails on sunny days - and there should also be equal number of heads as on not-sunny days. So we would have the same ratio in both cases (i.e. the numbers given in the challenge would have to be x% and (100-x)%.

That means that we either have insufficient trials to be meaningful, or the coin is not fair. So I don't think the data can support this conclusion.

Fyz

"the coin comes up heads 70% of the time on sunny days, this infers that 30% of the time it came up tails not 45%?"

No. Re-read the question. It says, "when the coin comes up heads, 70% of the time it's a sunny day". At this point, you are only looking at instances when the coin comes up heads. 70% of THOSE are sunny days, the other 30% are non-sunny days with heads, not tails on sunny days! Then he looks at instances where the coin comes up tails and finds that only 45% of THOSE are sunny days. The other 55% of tails tosses would then be on non-sunny days.

Where is this guy standing when he flips the coin.

If he is in Seattle, WA. He might have the same coin toss ratio only with mostly rainy days.

Might it have something to do with one side of the coin being heavier then the other side and the borometric presure being different with the air being less dense on sunny days.

There might be something a little more scientific rather then mathematical.

Sounds reasonable, well back to the drawing board. Computer crashed the other day so have to catch up.

Regards JD.

six one way, a half dozen the other.

He flips the coin n times of which H times heads and T times tails.

During these n times S times was sunny and N times not sunny.

The chance of flipping the coin on a sunny day heads up is H/nXS/n

S=0.7H+0.45T

If we consider that H/n=T/n=0.5 then the chance of flipping the coin heads up on a sunny day is 28.75% and on a non sunny day - 21.25%.By these consideration there is no difference whether flipping the coin on a sunny day or not, but rather the percentage of sunny and not sunny days.

The results Jack got were due to the fact, that when he flipped heads on, incidentally was a sunny day.

50:50 ratio of heads-to-tails on sunny days implies:

X heads on sunny days => X*3/7 heads on not-sunny days
X tails on sunny days => X*11/9 tails on not-sunny days

=> ratio of heads to tails on not-sunny days is 27:77

That means that the coin cannot be fair both on sunny and on not-sunny days

Fyz