No Calculator Problem

Reply #4

ash190 — Wed, 04 Oct 2023 14:28:32 GMT

good solutions, i should discover this forum when I was in high school

Reply #3

cottoneyedjoe — Tue, 12 Sep 2023 17:40:33 GMT

Good solution, and sorry to underwhelm you

Reply #2

Wavepack — Sun, 10 Sep 2023 19:30:49 GMT

Nice solution.

Reply #1

db101 — Sun, 10 Sep 2023 18:30:06 GMT

5p(1 - p)^4 = 10p^3(1 - p)^2

Right off the bat you can see p = 0 and p = 1 are two of solutions. Divide them out and you are left with

(1 - p)^2 = 2p^2

The only non-negative solution is p = sqrt(2) - 1.

Not much of a clever trick to be honest but it's kind of an interesting problem. Whenever I see a biased coin problem I always wonder how one could make a real coin with a specific bias. Maybe gluing together two disks of different densities? 3-D print one with asymmetric pockets... [ More ]

No Calculator Problem

cottoneyedjoe — Sun, 10 Sep 2023 14:46:07 GMT

This is a probability problem from a math competition where no calculators were allowed. The particulars were chosen so that a student could do it by hand. I thought I'd share it because there is a mildly clever trick involved.

An unfair coin has a probability p of landing on heads when it is flipped once. When the coin is flipped 5 times, the probability of getting exactly 1 head is equal to the probability of getting exactly 3 heads. What are all the possible values of p?

(Hint: There... [ More ]