Consider a vide game with two possible outcomes success or f

Consider a vide game with two possible outcomes: success or failure. You can play up to 10 rounds. In each round, if you succeed, you get a prize of $10, and if you fail, you get nothing. It costs $5 to enter the game in each round. You do not know the success probability but can assume that it is uniformly distributed in the interval [0,1]. Determine the optimal play strategy.

Solution

Probability of success =

P(s)

=x

Probability of failure =

P(f)

=1-x

if you succeed, you get a prize of $10

if you fail, you get $0

costs $5 to enter the game

Considering a success in the first round

expected value

=

$10x + 0(1-x)-5

=

10x-5

Let k successes maximize the profit giving optimum strategy

Probability of k successes in n trials (event S_k) = P(S_k) = (n k)*p^k *q^n-k

Here n = 10, ie P(S_k) = (10 k)*x^k *(1-x)^10-k

Expected value = 10* 10!/(k!*(10-k)! )*x^k *(1-x)^10-k – 50

Optimal strategy is to maximize the expected profit