Answer to Question #225191 in Statistics and Probability for Reyad

Question

A container has ‘55’ defective and 3 non defective items. if a sample of 2 items

are drawn one after another without replacement what is the probability that, the

sample will at most one defective

Answer 1

In this question:

We let the number of defective items in the sample $X$

$P\left(X=0\right)=\frac{\begin{pmatrix}3\\ 2\end{pmatrix}\begin{pmatrix}55\\ 0\end{pmatrix}}{\begin{pmatrix}55+3\\ 2\end{pmatrix}}=\frac{3\left(1\right)}{1653}$

$P\left(X=0\right)=\frac{1}{551}$

$P\left(X=1\right)=\frac{\begin{pmatrix}3\\ 1\end{pmatrix}\begin{pmatrix}55\\ 1\end{pmatrix}}{\begin{pmatrix}55+3\\ 2\end{pmatrix}}=\frac{3\left(55\right)}{1653}$

$P\left(X=1\right)=\frac{55}{551}$

$P\left(X\le 1\right)=P\left(X=0\right)+P\left(X=1\right)$

$P\left(X\le 1\right)=\frac{1}{551}+\frac{55}{551}$

$P\left(X\le \:1\right)\approx 0.1016$

Therefore, the probability that, the sample will at most one defective is 0.1016.