A standard roulette wheel in the U S has the numbers 1 throu

A standard roulette wheel in the U. S. has the numbers 1 through 36 plus 0 and 00. Of the numbers 1 through 36, 18 are colored red and 18 are colored black. The numbers 0 and 00 are colored green. If a person observes 20 consecutive spins of the wheel, what is the probability that the color red occurs 15 or fewer times? You can assume that results of separate spins are independent so that this probability can be modeled using the binomial distribution.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( red colour occured) = 18/38 = 0.4737
P( X < = 15) = P(X=15) + P(X=14) + P(X=13) +......P(X=0)
= ( 20 15 ) * 0.4737^15 * ( 1- 0.4737 ) ^5 + ( 20 14 ) * 0.4737^14 * ( 1- 0.4737 ) ^6 + ( 20 13 ) * 0.4737^13 * ( 1- 0.4737 ) ^7 + ( 20 12 ) * 0.4737^12 * ( 1- 0.4737 ) ^8 + .....( 20 0 ) * ( 0.4737^0) * ( 1 - 0.4737 )^20

= 0.997