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Bent Coin Probabilities (January 2024 Challenge Question)

Posted December 31, 2023 12:00 AM
Pathfinder Tags: challenge question

You and a friend have one cupcake. You decide to flip a coin to determine who gets the cupcake. The loser gets no cupcake at all.

However, the only coin you have is a coin with a bend in it.

How do you fairly flip the coin to determine who gets the cupcake? What are the minimal number of flips required to determine who wins?

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#1

Re: Bent Coin Probabilities (January 2024 Challenge Question)

12/31/2023 1:47 AM

You flip the coin in the air and the other guy calls it, 2 out of 3 wins...

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#2

Re: Bent Coin Probabilities (January 2024 Challenge Question)

12/31/2023 4:51 AM

You ask a third party, who has not seen the coin, to call for your friend, and, flip it only once.

If there are no other people around: you take two similar pieces of paper, write an H on one and a T on the other; fold them in half; shake them in your hands, and, get your friend to choose one to determine if he gets heads or tails prior to flipping the coin once.

No paper or writing implement: figure out a way to "draw straws" to see who gets heads, before flipping the coin only once.

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#3

Re: Bent Coin Probabilities (January 2024 Challenge Question)

12/31/2023 8:05 AM

Flip the coin until there is a head followed by a tail or a tail followed by a head. If HT you win, TH your friend wins.

The coin has no memory, so the sequence HT and TH have the same probability, the probability of heads x the probability of tails.

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#4
In reply to #3

Re: Bent Coin Probabilities (January 2024 Challenge Question)

12/31/2023 12:11 PM

GA: but I think you need to reword it a bit.

The way you've written it the first toss will determine the result.

I think you need to repeatedly toss the coin twice until you get an HT or a TH.

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#5
In reply to #4

Re: Bent Coin Probabilities (January 2024 Challenge Question)

12/31/2023 5:46 PM

Thanks, you are correct. You need fresh tosses for each trial. Otherwise, the coin does have memory.

Example, suppose heads probability is 0.7, tails is 0.3.

2 tosses of coin, probabilities:

HH=0.49, HT=0.21, TH=0.21, TT=0.09

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#18
In reply to #4

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/30/2024 1:49 PM

Good answer for rephrasing the answer...in the form of an answer.

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#17
In reply to #3

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/30/2024 1:48 PM

Admin chiming in...this answer correctly identifies how to do this in two flips.

Over the course of two flips: there are 4 possible outcomes.

  • Heads-Heads
  • Heads-Tails
  • Tails-Heads
  • Tails-Tails

If the coin favors one side, we know that any heads-heads or tails-tails results are biased. So the flippers need to guess based on sequence, not side.

All HH or TT flips need to be disregarded. The minimum number of flips needed is 2.

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#6

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/02/2024 12:30 AM

Tell your opponent that if it lands on its edge he wins, I would do anything for a cupcake! Only takes 1 throw to decide who wins and who is the crumby loser!

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#7

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/02/2024 2:22 PM

Keep alternately flipping coin till the first person gets the least likely side up, ie the concave side down, concave side up

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#10
In reply to #7

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/05/2024 8:08 AM

Good idea: let's see.

Suppose that the probability of the more likely flip result is p, and, the probability of the first player winning is A. Then

A = (1 - p) + p2(1 - p) + p4(1 - p) ........

A = (1 - p)(1 + p2 + p4 + ......) (1)

Ap2 = (1 - p)( p2 + p4 + ......) (2)

(1) - (2)

A(1 - p2) = (1 - p)

A = 1/(1 + p)

Similarly if B is the probability of the second player winning:

B = p/(1+p)

CHECK A + B = (1+p)/(1 + p) = 1

For example if p =2/3 then A = 3/5 and B = 2/5

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#8

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/04/2024 7:16 PM

Sneak a little jelly on one side before flipping,and call that side.The sticky side always lands down.

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#9
In reply to #8

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/04/2024 8:40 PM

Dropped toast usually lands on the floor butter-side-down.

"A study at Manchester Metropolitan University involving dropping 100 slices under laboratory conditions established that toast typically lands on the floor butter-side-down as a result of the manner in which it is typically dropped from a table, and the aerodynamic drag caused by the air pockets within the bread. The toast is typically butter-side-up when dropped. As it falls, it rotates; given the typical speed of rotation and the typical height of a table, a slice of toast that began butter-side-up on the table lands butter-side-down on the floor in 81% of cases."

https://en.wikipedia.org/wiki/Buttered_cat_paradox#:~:text=A%20study%20at%20Manchester%20Metropolitan,air%20pockets%20within%20the%20bread.

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#11
In reply to #9

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/05/2024 10:15 AM

I wrote a long story based on the idea that toast always lands peanut- butter side down,and cats always land on their feet.I wrote in back in the 1980's and called it The CAT Toast Generator.

It consisted of a cat with a peanut butter toast strapped to it's back,placed on a pivot that rotated because each attachment was trying to get to the bottom.

I copyrighted it ,but never published it.

However,I did show it to friends after I Copyrighted it,and they got a kick out of it.

One of my friends moved to Australia and a few months later, an Aussie published something almost exactly the same on the net.

Plagiarism is not restricted to one country it seems.

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#15
In reply to #11

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/10/2024 12:36 PM

This sounds like the 'cat with butter on both sides' levitation postulate.

The rheology of cats might amuse.

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#12

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/05/2024 4:48 PM

Let your friend flip the coin. While it’s in the air, grab the cupcake and run. All is fair when dessert is on the line.

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#13

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/05/2024 7:14 PM

Just flatten the coin with this handy press....

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#21
In reply to #13

Re: Bent Coin Probabilities (January 2024 Challenge Question)

02/06/2024 7:23 AM

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Re: Bent Coin Probabilities (January 2024 Challenge Question)

06/01/2024 7:04 PM

Most essential item in any toolbox.

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#14

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/09/2024 7:11 AM

One flip.

I flip the coin (since this is stated in the Challenge), and under a table I take the coin in one hand, keeping my hands out of sight but widely separated to avoid lack-of-trust-induced violent conflicts. The cupcake-contender chooses a hand: left or right. Do not forget to say "my left" or "your left" - merely pointing would avoid ambiguities. If he chooses the hand with the coin in it, he's the lucky one.

So, one flip is all it takes.

Common practice in table tennis where tossing bats is not-done, and the chance of the available ball landing on one side is 1 and on the other side is 0 :-)

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#16

Re: Bent Coin Probabilities (January 2024 Challenge Question)

01/10/2024 12:42 PM

The degree of bend is not stated. 180o (fully bent in half)?

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#20
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Re: Bent Coin Probabilities (January 2024 Challenge Question)

02/06/2024 7:20 AM

If it were <...180o (fully bent in half)...> then it would flip heads every time or tails every time, depending on which way it had been <...bent...>.

Which would be pointless, really.

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#19

Re: Bent Coin Probabilities (January 2024 Challenge Question)

02/06/2024 5:21 AM

So, cutting the <...cupcake...> into two pieces, though fair, is not viable?

That sort of thing can leave a bad taste in the mouth.

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