Question

A box contains 16 transistors, 3 of which are defective. If 3 are selected at random, find the probability of the statements below.a. All are defectiveb. None are defectiveWith, and without replacement

Answer 1

Given 16 transistors, 3 of which are defective, 13 are not defective

Part A: All are defective.

With replacement

$\begin{gathered} P=\frac{3}{16}\cdot\frac{3}{16}\cdot\frac{3}{16} \\ P=\frac{27}{4096} \\ \\ \text{The probability that all are defective with replacement is }\frac{27}{4096} \end{gathered}$

Without replacement

$\begin{gathered} P=\frac{3}{16}\cdot\frac{2}{15}\cdot\frac{1}{14} \\ P=\frac{143}{280} \\ \\ \text{The probability that all are defective without replacement is }\frac{1}{560} \end{gathered}$

Part B: None are defective

With replacement

$\begin{gathered} P=\frac{13}{16}\cdot\frac{13}{16}\cdot\frac{13}{16} \\ P=\frac{2197}{4096} \\ \\ \text{Therefore, the probability that none are defective with replacement is }\frac{2197}{4096} \end{gathered}$

Without replacement

$\begin{gathered} P=\frac{13}{16}\cdot\frac{12}{15}\cdot\frac{11}{14} \\ P=\frac{143}{280} \\ \\ \text{Therefore, the probability that none are defective without replacement is }\frac{143}{280} \end{gathered}$