This is problem 8.11 in the Marcel Finan book.
A store has 80 modems in its inventory, 30 coming from Source A and the remainder from Source B. Of the modems from Source A, 20% are defective. Of the modems from Source B, 8% are defective. Calculate the probability that exactly two out of a random sample of five modems from the store’s inventory are defective.
$$P(\text{defective}) = \frac{3}{8} \frac{1}{5} + \frac{5}{8} \frac{8}{100}=\frac{1}{8}$$
$$P(\text{exactly 2 defective}) = \binom{5}{2} P(\text{defective})^2 (1-P(\text{defective}))^3 = 0.104675$$
But the answer key says 0.102. What have I done wrong?
Best Answer
Consider all modems in the store totalling $80$.
Once they are comingled, it doesn't matter where they come from.
$$\begin{array}\\ \hline\ &\text{Source} &A &B &&\text{Total}\\ \hline &\text{Total} &30 &50 &&80\\ &\% \text { Defective} &20\% &8\%\\ &\text{Defective} &\;\ 6 &\;\ 4 &&10\\ &\text{Not Defective} &24 &46 &&70\\ \hline \end{array}$$
Total number defective = $10$.
Total number non-defective = $70$.
Choose $5$ modems.
Probability of getting exactly $2$ defective is:
$$\frac{\binom {10}2\binom {70}3}{\binom {80}5}=0.102\qquad\blacksquare$$