Question

A  shipment  of  7  television  sets  contains  2  defective  sets.   A  hotel  makes  a  random purchase of 3 of the sets.  If x is the number of defective sets purchased by the hotel, find the probability distribution of X.  Express the results graphically as a probability histogram.

Answer 1

The probability that there will be 0 defective sets:

$p(0) = \frac{{C_5^3}}{{C_7^3}} = \frac{{5!}}{{3!2!}} \cdot \frac{{3!4!}}{{7!}} = \frac{{3 \cdot 4 \cdot 5}}{{5 \cdot 6 \cdot 7}} = \frac{2}{7}$

1 defective sets:

$p(1) = \frac{{C_5^2C_2^1}}{{C_7^3}} = 2 \cdot \frac{{5!}}{{3!2!}} \cdot \frac{{3!4!}}{{7!}} = 2 \cdot \frac{{3 \cdot 4 \cdot 5}}{{5 \cdot 6 \cdot 7}} = \frac{4}{7}$

2 defective sets:

$p(2) = \frac{{C_5^1C_2^2}}{{C_7^3}} = 5 \cdot \frac{{3!4!}}{{7!}} = 5 \cdot \frac{6}{{5 \cdot 6 \cdot 7}} = \frac{1}{7}$

We have  the probability distribution:

$\begin{matrix} X&0&1&2\\ p&{\frac{2}{7}}&{\frac{4}{7}}&{\frac{1}{7}} \end{matrix}$

 Probability histogram: