A student of mine was asked this question in an interview Co

A student of mine was asked this question in an interview. \"Consider the following game: A cup is filled with 100 pennies. The cup in shaken, and the pennies are poured onto a table. If at least 60 of the pennies are Heads, you win $20, Otherwise, you lose $1. Is this a good game to play?\" (That is does it have a positive expected value?). Answer this using simulation in R. a) Determine how much you would win or lose. b) Does it appear to be a fair game?

Solution

Calculating this many pennies with the binomial is a pain,
which is why we\'d use a normal approximation. Here,
we let µ=n*p = 100*.5 = 50 (the expected number of pennies to end up heads)
and s = v(n*p*q) = v(100*.5*.5) = 5

We can find a Z score, noting that above 60 heads is what peaks our interest.

Z = (60-50)/5 = 2

Noting on our Z-score chart that the area to the right of Z = 2 is 0.0228,
this means we have a probability of 2.28% to land 60 or more heads.
Thus, the expected earnings from the game is 0.0228*($20) - 0.9772*($1) = -0.5212
which means this is a bad game to be playing