Sikhism a religion founded in the 15th century in India is g

Sikhism, a religion founded in the 15th century in India, is going through turmoil due to a rapid decline in the number of Sikh youths who wear turbans (Washington Post, March 29, 2009). The tedious task of combing and tying up long hair and a desire to assimilate has led to approximately 25% of Sikh youths giving up the turban. a. What is the probability that exactly two in a random sample of five Sikh youths wear a turban? b. What is the probability that two or more in a random sample of five Sikh youths wear a turban? c. What is the probability that more than the expected number of Sikh youths wear a turban in a random sample of five Sikh youths? d. What is the probability that more than the expected number of Sikh youths wear a turban in a random sample of 10 Sikh youths?

Solution

Wearing a turban = 0.75 ; q = 1-p = 0.25

P(X=r) = nCr * p^r * q^(n-r)

a) n=5, r = 2

P(X=2) = 5C2 * 0.75^2 * 0.25^3 = 0.0879

b) P(X>=2) = 1 - P(X<2) = 1 - P(X=0) - P(X=1)

P(X=0) = 0.000976

P(X=1) = 0.014648

Req P = 0.9844

c) E(X) = np = 5*0.75 = 3.75

P = P(X=4) + P(X=5) = 0.3955 + 0.2373 = 0.6328

d) E(X) = 10*.75 = 7.5

P = P(X=8) + P(X=9) +P(X=10) = 0.281567 + 0.1877 + 0.05631 = 0.5256