Chapter 9, Problem 18CT

To determine

To find: probability that your favorite song is one of the 20 that you hear that night.

Answer to Problem 18CT

  $\frac{1}{15}$

Explanation of Solution

Given:

Total number of songs = 300

Total number of chosen songs = 20

Total number of favorite songs = 1

Total number of ways to choose 20 songs out of 300 songs is:

  $n\left(S\right)=C{}_{20}$

Let E be the event that your favorite song is one of the 20 that you hear that night.

The favorite song can be chosen in $C{}_1$ way.

And the remaining 19 songs out of remaining 299 songs can be chosen in $C{}_{19}$ ways

So,

  $n\left(E\right)=C{}_1\times C{}_{19}$

Thus, probability that your favorite song is one of the 20 that you hear that night is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{C{}_1\times C{}_{19}}{C{}_{20}}\\ =\frac{1\times \frac{299\text{!}}{280\text{!}\times 19\text{!}}}{\frac{300\text{!}}{280\text{!}\times 20\text{!}}}\end{array}$

  $\begin{array}{c}=\frac{20}{300}\\ =\frac{1}{15}\end{array}$