Independent solution

How to solve this Binomial Distribution question

Setup

Setup

A couple counts as a success only when both of its members survive. Independence turns that intersection into a product.

q=Pr⁡(both members survive)=(0.79)2=0.6241q=\Pr(\text{both members survive})=(0.79)^2=0.6241

Model

Model

Let N count successful couples. Independence across all people makes the seven couple indicators independent Bernoulli variables.

N∼Binomial⁡(7,0.6241)N\sim\operatorname{Binomial}(7,0.6241)
Pr⁡(N≥3)=1−Pr⁡(N≤2)\Pr(N\ge3)=1-\Pr(N\le2)

Compute

Compute

Evaluate the short lower tail and take its complement.

Pr⁡(N≤2)=∑k=02(7k)(0.6241)k(0.3759)7−k\Pr(N\le2)=\sum_{k=0}^{2}\binom{7}{k}(0.6241)^k(0.3759)^{7-k}
Pr⁡(N≤2)=0.07477428837…\Pr(N\le2)=0.07477428837\ldots
Pr⁡(N≥3)=0.9252257116…\Pr(N\ge3)=0.9252257116\ldots

Answer

Answer

The requested probability rounds to 0.925.

Pr⁡(N≥3)≈0.925(D)\boxed{\Pr(N\ge3)\approx0.925\quad\text{(D)}}