Question

• Use the binomial distribution to determine which of the following three games has the highest probability of winning. What is the minimum rational price you should charge to play this game if you must pay a winner $5.00? Game 1: throw a single die 6 times. You win if you roll a 1 at least once. Game 2: throw a single die 12 times. You win if you roll a 1 at least twice. Game 3: throw a single die 18 times. You win if you roll a 1 at least three times.

Answer 1

Game 1:

Probability of winning here is computed as:

= Probability of getting 1 at least once in 6 dice throws

= 1 - Probability of not getting a one even once

= 1 - (5/6)⁶

= 0.6651

Game 2:

Probability of getting 1 at least twice in 12 rolls is computed here as:

= 1 - Probability of not getting a 1 even once - Probability of getting a 1 exactly once

We compute the above probabilities using binomial distribution function as:

-1- (5/0) - (12)(1/6)(5/6)"

= 0.6187

Game 3:

Probability of getting 1 at least thrice in 18 rolls is computed here as:

= 1 - Probability of not getting a 1 even once - Probability of getting a 1 exactly once - Probability of getting a 1 exactly twice

We compute the above probabilities using binomial distribution function as:

-1- (5/0)s – (18)(1/6)(5/6)r - (18)(1/6)°(5/6)ས

=0.973

Therefore Game 1 clearly has the highest probability of winning here.

For paying the game winner $5, the price that should be charged to the winner is computed by making the expected value of the game to be 0 to make it a fair game. Let the price charged be $P

Then, we have here:

P*(Probability of not winning ) = 5*Probability of winning

P(1 - 0.6651) = 5*0.6651

P = $9.9298

Therefore the fair price here is given as: $9.9298