Question

Binomial Distribution. Surveys repeatedly show that about 40% of adults in the U.S. indicate that if they only had one child, they would prefer it to be a boy. Suppose we took a random sample of 15 adults. Based on a proportion of .40, what is the exact probability of X < 7.

Answer 1

Solution

Given that ,

p = 0.40

1 - p = 1 - 0.40 = 0.60

n = 15

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

P(X < 7) = ((15! / 0! (15 - 0)!) * 0.40⁰ * (0.60)^{15 - 0} + ((15! / 1! (15 - 1)!) * 0.40¹ * (0.60)^{15 - 1} + ((15! / 2! (15 - 2)!) * 0.40² * (0.60)^{15 - 2} + ((15! / 3! (15 - 3)!) * 0.40³ * (0.60)^{15 - 3} +((15! / 4! (15 - 4)!) * 0.40⁴ * (0.60)^{15 - 4} + ((15! / 5! (15 - 5)!) * 0.40⁵ * (0.60)^{15 - 5} + ((15! / 6! (15 - 6)!) * 0.40⁶ * (0.60)^{15 - 6}

=  0.0005 + 0.0047 + 0.0219 + 0.0634 + 0.1268 + 0.1859 + 0.2066

Probability = 0.6098