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?

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