binomial probability distribution brand recognition

Question

The brand name of Miss Fields has a 90% recognition rate. If Miss Fields herself wants to verify that rate by beginning with a small sample of 10 randomly selected consumers,

a) find the probability that all BUT 10 consumers recognize her brand

b) also find the probability that the number of consumers recognize her brand name is exactly 9

Answer 1

p = 0.90 = the probability that a randomly selected individual recognizes Miss Fields brand

n = 10 = number of randomly selected individuals

X = random variable that represents the number of individuals out of n=10 that recongize Miss Fields

a) What we want to find is Pr(X < 10). It's much easier to use the complement rule, which states for an event A, and its complement A^c:

Pr(A) + Pr(A^c) = 1

Pr(A) = 1 - Pr(A^c)

If A = {X < 10}, then A^c= {X ≥ 10}

Since the maximum number for X is 10, then A^c= {X = 10}

Hence

Pr(X < 10) = 1 - Pr(X=10)

We use the binomial distribution to find Pr(X=10)

Pr(X=10) = [10(Choose)10] * (0.9)^10 * (0.1)^0

= 0.34868

Pr(X < 10) = 1 - 0.34868 = 0.65132

b) We use the binomial distribution to find Pr(X=9)

Pr(X=9) = [10(Choose)9] * (0.9)^9 * (0.1)^1 = 0.38742