Chapter CSB, Problem 4.12E

To determine

Find the probability $P\left(4girls,2boysor4boys,2girls\right).$

Answer to Problem 4.12E

  $\frac{70}{143}.$

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:

Combination is an arrangement or selection of objects in which order is not important.

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 $3003$.

Calculate the probability that the salesperson will help $4$ girls and $2$ boys as shown below,

Possible number of outcomes when $4$ girls are chosen from $6$ girls is ${}_6{C}_4.$

Possible number of outcomes when $2$ boys are chosen from $8$ boys is ${}_8{C}_2.$

So, total number of possible outcomes is,

  $\begin{array}{l}{}_6{C}_4{\cdot }_8{C}_2=\frac{6!}{4!\left(2!\right)}\cdot \frac{8!}{2!\left(6!\right)}\\ =15\cdot 28\\ =420\end{array}$

The required probability is given by,

  $\begin{array}{l}Probability=\frac{possiblenumberofoutcomes}{totalnumberofoutcomes}\\ =\frac{420}{3003}\\ =\frac{20}{143}\\ So,P\left(4girls,2boys\right)=\frac{20}{143}.\end{array}$

Calculate the probability that the salesperson will help $2$ girls and $4$ boys as shown below,

Possible number of outcomes when $2$ girls are chosen from $6$ girls is ${}_6{C}_2.$

Possible number of outcomes when $4$ boys are chosen from $8$ boys is ${}_8{C}_4.$

So, total number of possible outcomes is,

  $\begin{array}{l}{}_6{C}_2{\cdot }_8{C}_4=\frac{6!}{2!\left(4!\right)}\cdot \frac{8!}{4!\left(4!\right)}\\ =15\cdot 70\\ =1050\end{array}$

The required probability is given by,

  $\begin{array}{l}Probability=\frac{possiblenumberofoutcomes}{totalnumberofoutcomes}\\ =\frac{1050}{3003}\\ =\frac{50}{143}\\ So,P\left(4boys,2girls\right)=\frac{50}{143}.\end{array}$

The probability that the salespeople will help $4$ girls and $2$ boys or $2$ girls and $4$ boys is,

  $\begin{array}{l}P\left(4girls,2boysor4boys,2girls\right)=P\left(4girls,2boys\right)+P\left(4boys,2girls\right)\\ =\frac{20}{143}+\frac{50}{143}\\ =\frac{70}{143}\end{array}$

Hence, $P\left(4girls,2boysor4boys,2girls\right)is\frac{70}{143}.$