Chapter CSB, Problem 4.14E

To determine

Find the probability $P\left(allgirlsorallboys\right).$

Answer to Problem 4.14E

  $0.009657$.

Explanation of Solution

Given information:

In a video store, $6$ girls and $8$ boys enter at the same time.

Number of available salespeople is $6.$

Calculation:

As there are $6$ girls and $8$ boys, then the total number of choices are $14$ out of which $6$ persons are chosen.

Calculate the total number of outcomes as,

  $\begin{array}{l}{}_{14}{C}_6=\frac{14!}{6!\left(8!\right)}\\ =3003\end{array}$

So, the total number of outcomes is $3003$.

The probability that salespeople will first help all boys is $C{}_6.$

The required probability is given by,

  $\begin{array}{l}P\left(allboys\right)=\frac{{}_8{C}_6}{{}_{14}{C}_6}\\ =\frac{\frac{8!}{6!\left(2!\right)}}{3003}\\ =\frac{28}{3003}\end{array}$

The probability that salespeople will first help all girls is $C{}_6.$

The required probability is given by,

  $\begin{array}{l}P\left(allgirls\right)=\frac{{}_6{C}_6}{{}_{14}{C}_6}\\ =\frac{\frac{6!}{6!\left(0!\right)}}{3003}\\ =\frac{1}{3003}\end{array}$

The probability that the salespeople will help all boys or all girls is,

  $\begin{array}{l}P\left(allboysorallgirls\right)=P\left(allboys\right)+P\left(allgirls\right)\\ =\frac{28}{3003}+\frac{1}{3003}\\ =\frac{29}{1001}\\ =0.009657\end{array}$

Hence, the probability $P\left(allboysorallgirls\right)is\mathrm{0.009657.}$