Problem of the Week #62: Monday September 9th, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by this twitter post.
If the possible fairness of a coin is a continuum, the probability of the coin being exactly fair (50H/50T) is 0. To begin exploring, let’s say the coin can only be rigged heads (100H/0T), fair (50H/50T), or rigged tails (0H/100T).
a). If we flip the coin once and get a head, what’s the probability that it’s a fair coin?
b). If we flip the coin once and get a head, what’s the probability of it landing on a head on the next flip?
c). If we flip the coin twice and get two heads, what’s the probability that it’s a fair coin? What’s the probability of it landing on a head on the next flip?
d). Let’s add two more possible fairness to the coin: ¾ head (75H/25T), and ¾ tails (25H/75T). Now if we flip the coin twice and get two heads, what’s the probability of it landing on a head the next flip?
e). With the same 5 possible fairness states, if we get n head on the first n flip, what’s the probability of it landing on a head the next flip?
f). If we keep adding more possible fairness states on the continuum and getting n heads on the first n flips, what’s the probability of it landing on a head the next flip?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.