Independent solution

How to solve this Binomial Distribution question

Setup

Setup

For one group, at least nine completers means exactly nine or exactly ten completers in a binomial count.

q=(109)(0.8)9(0.2)+(0.8)10=0.375810q=\binom{10}{9}(0.8)^9(0.2)+(0.8)^{10}=0.375810

Model

Model

The two groups are independent. Exactly one group meets the threshold through either of two disjoint orders: the first succeeds and second fails, or vice versa.

Pr⁡(exactly one group)=q(1−q)+(1−q)q\Pr(\text{exactly one group})=q(1-q)+(1-q)q

Compute

Compute

The one-group probability is approximately 0.375810. Doubling the product of this probability and its complement gives approximately 0.469154.

2q(1−q)=0.4691542q(1-q)=0.469154

Answer

Answer

The probability that exactly one group meets the completion threshold is approximately 0.469, selecting choice E.

0.469(E)\boxed{0.469\quad\text{(E)}}