Question

The American roulette wheel has 38 total slots of which 18 are colored red, 18 are colored black, and 2 are colored green. The red and the black slots are numbered 1 through 36. The green slots are numbered 0 and 00.
Construct the probability distribution for betting $1 on the second dozen numbers, 13 through 24, (which pays 2 to 1) in the game of roulette and calculate the corresponding mean and standard deviation.

Answer 1

solution:

From the given information

No.of red slots = 18

No.of black slots = 18

No.of green slots = 2 { 0 , 00 }

Total no.of slots = 38

No.of favourable slots to win = 12 { 13 through 24 }

No.of nonfavourable slots = 38-12 = 26

Let X be the random variable representing amount gain or lose by the player

The probability distribution of X is

X	2	-1
P(X)	12/38	26/38

Here, The expected value or mean of X= E{X] =

= (2 * 12/38) + (-1 * 26/38)

= 24/38 - 26/38

= -2/38

= -$0.0526

Therefore, Mean of X = -$ 0.0526

Variance of X () = E[X^2] - E[X]^2

=   - ( ) ^2

= (4 * 12/38 + 1 * 26/38) - (-2/38)^2

= 74/38 - (-2/38)^2

= 1.945

Therefore, standard deviation () = =   =~ $ 1.395