Question

A casino wants to offer a new game with (potentially) spectacular prizes:The player flips a fair coin until they flip heads.The starting prize is \$2$2 (if the player flips heads on the first flip).The prize doubles with each flip after the first.For example, if somebody flipped tails-tails-tails-heads, then their prize would be \$16$16 (\$2($2 doubled three times).). In other words, the prize is \$2^n,$2 n , where nn is the number of flips it took for the player to flip heads for the first time.How much should the casino charge to play this game?

Answer 1

The casino should charge for this game at least $1 to break even.

Explanation:

We can define the prize function as

`M(n)=2^( n+1)`

where M is the prize money and n is the number of tails in continous flips.

The probability of n consecutive tails can be calculated as
`p^n=0.5^n`. The probaility of getting a head after the n consecutive tails is
`p=0.5`, so the probability of having n consecutive tails and a head is
`p^(n+1)=0.5^(n+1)`

Then we can calculate the expected value of M as

`E(M)=p_i*M_i=(0.5)^(n+1)*2^(n+1)=1^(n+1)=1`

The expected money prize for this game is $1, so the casino should charge to play at least $1 to break even.