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A roulette wheel is numbered from 0 to 36. 0 is Green. Half of numbers are

Red and half are Black. The game has an entrance fee $1. The player then stakes

$10 and must choose the parity (Odd or Even) and the color (Red or Black). If

he gets right parity or color $12 is returned, that is a gain is $1. If he get right

both parity and color $20 is returned, , that is a gain is $9. If he does guesses

neither correct color nor parity, and the number is not 0, then the entrance fee $1

is returned. If 0 comes up, the player gets nothing.

(a) If X is the gain on a single game, complete the table of the probability distribution

of random variable X:

(b) Find E(X) and standard deviation of X

(c) If player plays twice, what is the probability that he comes out ahead (i.e.

positive net gain).

(d) If player plays this game fifty times, find the mean and standard deviation of

his overall net gain.

(e) Use your answer to part (d) and a suitable approximation to calculate the

probability of coming out ahead after playing fifty games.

(f) Given a roulette wheel where the half of odd numbers are Red and half are

Black, and similarly for even numbers, check that color and parity appear

independently.

A roulette wheel is numbered from 0 to 36. 0 is Green. Half of numbers are

Red and half are Black. The game has an entrance fee $1. The player then stakes

$10 and must choose the parity (Odd or Even) and the color (Red or Black). If

he gets right parity or color $12 is returned, that is a gain is $1. If he get right

both parity and color $20 is returned, , that is a gain is $9. If he does guesses

neither correct color nor parity, and the number is not 0, then the entrance fee $1

is returned. If 0 comes up, the player gets nothing.

(a) If X is the gain on a single game, complete the table of the probability distribution

of random variable X:

(b) Find E(X) and standard deviation of X

(c) If player plays twice, what is the probability that he comes out ahead (i.e.

positive net gain).

(d) If player plays this game fifty times, find the mean and standard deviation of

his overall net gain.

(e) Use your answer to part (d) and a suitable approximation to calculate the

probability of coming out ahead after playing fifty games.

(f) Given a roulette wheel where the half of odd numbers are Red and half are

Black, and similarly for even numbers, check that color and parity appear

independently.