# What’s Past is Not Prologue - coin flip question

A question posed in 538 Riddler and the article "What’s Past is Not Prologue" available at SSRN: https://ssrn.com/abstract=3034686 is: "You are presented with two coins: one is fair, and the other has a 60% chance of coming up heads. Unfortunately, you don't know which is which. How many flips would you need to perform in parallel on the two coins to give yourself a 95% chance of correctly identifying the biased one?"

The solution provided is 143. However, I believe the correct answer is 134.

After 134 flips, using the joint distribution binomial formula provided in the paper there is a 94.40% probability that the biased coin will land heads up more times than the fair coin, 4.34% probability that the fair coin will land heads up more times than the biased coin and 1.26% probability that the two will land heads up an equal number of times.

If you choose the coin that lands heads up more and guess randomly in cases where they land heads up an equal number of times, after 134 flips you will have 94.40% + 50%*1.26% = 95.03% probability of identifying the biased coin.

Which answer is the correct, 143 or 134?

## 1 Answer

Your method works well and I think you are correct

The following R code confirms your calculations

signdiff <- function(x,y){sign(x-y)}
probdecide <- function(flips, prob1, prob2){
signmat <- outer(0:flips, 0:flips, signdiff)
probmat <- dbinom(0:flips,flips,prob1) %*% t(dbinom(0:flips,flips,prob2))
c(sum(probmat[signmat == 1]),
sum(probmat[signmat == 1]) + 0.5*sum(probmat[signmat == 0]) )
}


giving

> probdecide(133, 0.6, 0.5)
[1] 0.9433231 0.9496884
> probdecide(134, 0.6, 0.5)
[1] 0.9440474 0.9503250
> probdecide(142, 0.6, 0.5)
[1] 0.9494897 0.9551124
> probdecide(143, 0.6, 0.5)
[1] 0.9501282 0.9556747


So $134$ is the smallest number of flips to give at least a $95\%$ chance of being correct if guessing randomly when both coins show the same number of heads, while $143$ is the smallest number of flips to give at least a $95\%$ chance of being correct if refusing to guess when both coins show the same number.