1 out of 3 people in the united states will develop shingles

1 out of 3 people in the united states will develop shingles during their lifetime. Imagine that you are a doctor who is about to examine 10 unrelated adults selected at random and before you examine them we ask you what are the chances that. a. At least one of them will get shingles? Explain your method. b. Exactly one of them will get shingles? Explain your method. c. At most one of them will get shingles? Explain your method. Round off each answers in each part of this question to three decimal places

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 = 1 out of 3 people in the united states will develop shingles = 1/3 = 0.333

a.
P( X < 1) = P(X=0)
= ( 10 0 ) * 0.333^0 * ( 1- 0.333 ) ^10
= 0.0174
P( X > = 1 ) = 1 - P( X < 1) = 0.983

b.
P( X = 1 ) = ( 10 1 ) * ( 0.333^1) * ( 1 - 0.333 )^9
= 0.087

c.
P( X < = 1) = P(X=1) + P(X=0)   
= ( 10 1 ) * 0.333^1 * ( 1- 0.333 ) ^9 + ( 10 0 ) * 0.333^0 * ( 1- 0.333 ) ^10
= 0.104