According to the ALA 90 of adult smokers started smoking bef

According to the ALA, 90% of adult smokers started smoking before turning 21 years old. Ten smokers 21 years old or older are randomly selected, and the number of smokers who started smoking before 21 is recorded. Explain why this is a binomial distribution. Find and interpret the probability that exactly 8 of them started smoking before the age of 21. Find and interpret the probability that at fewer than 8 of them started smoking before the age of 21. Find and interpret the probability that between 7 and 9 of them, inclusive, started smoking before the age of 21.

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

a)
It is binomial because only 2 outcomes are possible

b)
P( X = 8 ) = ( 10 8 ) * ( 0.9^8) * ( 1 - 0.9 )^2
= 0.19371
c)
P( X < 8) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 7 ) * 0.9^7 * ( 1- 0.9 ) ^3 + ( 10 6 ) * 0.9^6 * ( 1- 0.9 ) ^4 + ( 10 5 ) * 0.9^5 * ( 1- 0.9 ) ^5 + ( 10 4 ) * 0.9^4 * ( 1- 0.9 ) ^6 + ( 10 3 ) * 0.9^3 * ( 1- 0.9 ) ^7 + ( 10 2 ) * 0.9^2 * ( 1- 0.9 ) ^8 + ( 10 1 ) * 0.9^1 * ( 1- 0.9 ) ^9 + ( 10 0 ) * 0.9^0 * ( 1- 0.9 ) ^10
= 0.070191
d)
P( X = 7 ) = ( 10 7 ) * ( 0.9^7) * ( 1 - 0.9 )^3
= 0.057396
P( X = 9 ) = ( 10 9 ) * ( 0.9^9) * ( 1 - 0.9 )^1
= 0.38742
P( 7 <= X <= 9) = 0.057396 + 0.38742 + 0.070191 = 0.515007