Question

Joe Smith plays a betting game with his friends. He puts in one dollar to join the game. If he wins the game, he gets $2. If he loses, he gets nothing, so his gain is –$1. Joe Smith is planning to play the game 100 times. The 100 outcomes will be independent of one another. Let V_i, i = 1, 2, ..., 100 be the 100 outcomes. Then each V_i has the same probability distribution:

v	-1	1
p(v)	0.6	0.4

a.) What is his expected total gain if he plays 100 times?

b.) What is the variance of his total gain?

c.) Use the Central Limit Theorem to compute the probability that his total gain will be positive. Hint: the CLT is stated in terms of the mean.

Answer 1

a) For each trial, E(V) = -1*0.6 + 1*0.4 = -0.2

E(V²) = 1

Therefore Var(V) = E(V²) - [E(V)]² = 1 - 0.2² = 0.96

Expected total gain = 100E(V) = 100*(-0.2) = -20

Therefore expected total gain here for 100 trials is -$20 that is a total loss of $20

b) Var(100V) = 100² Var(V) = 100² *0.96 = 9600

Therefore 9600 is the required variance of his total gain here.

c) Using central limit theorem, the distribution here could be approximated as:

The required probability here is computed as:

Converting it to a standard normal variable, we get:

Getting it from the standard normal tables, we get:

Therefore 0.4191 is the required probability here.