Bent Coin Probabilities (January 2024 Challenge Question)

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

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.

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

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

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.

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!

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

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

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

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.

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.

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

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 :-)

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

If it were <...180^o (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.

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

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