Chapter 10.2, Problem 20E

(a)

To determine

To calculate:

The probability value of the selection of three red marbles when every marble is gets replaced before the selection

Answer to Problem 20E

The probability value of the selection of three red marbles when every marble is gets replaced before the selection is $0.091125$ .

Explanation of Solution

Given information:

Number of red marble: $n_R=9$

Number of yellow marbles: $n_y=7$

Number of blue marbles: $n_b=4$

Calculation:

Here, total numbers of marbles in bag is calculated as:

  $n=n_R+n_y+n_b=9+4+7=20$

In the first draw, need to select the three marbles of red colored when every marble is getting replaced before the selection.

Consider this event as event $P\left(A\right)=P\left(3R\right)$ .

Probability values of events $A$ of three red colored marbles selection is calculated as:

  $\begin{array}{l}P\left(A\right)=P\left(3R\right)\\ P\left(A\right)=\frac{9}{20}\times \frac{9}{20}\times \frac{9}{20}\\ P\left(A\right)=\frac{729}{8000}\\ P\left(A\right)=0.091125\end{array}$

(b)

To determine

To calculate:

The probability value of the selection of three red marbles when every marble is not replaced before the selection of the next marble

Answer to Problem 20E

The probability value of the selection of three red marbles when every marble is not replaced before the selection of the next marble is $0.0737$ .

Explanation of Solution

Given information:

Number of red marble: $n_R=9$

Number of yellow marbles: $n_y=7$

Number of blue marbles: $n_b=4$

Calculation:

Here, total numbers of marbles in bag is calculated as:

  $n=n_R+n_y+n_b=9+4+7=20$

In the draw, need to select the three marbles of red colored when every marble is not replaced before the selection of the next marble.

Consider this event as event $P\left(A\right)=P\left(3R\right)$ .

Here, in the first draw nine marbles of red colored selected, then next one marble is not replaced and there is remaining of eight red colored marbles. Now, need to select one red marble again without replacement that means there is remaining of seven red colored marbles.

Probability values of events $A$ of three red colored marbles selection without replacement are calculated as:

  $\begin{array}{l}P\left(A\right)=P\left(3R\right)\\ P\left(A\right)=\frac{9}{20}\times \frac{8}{20}\times \frac{7}{20}\\ P\left(A\right)=\frac{504}{8000}\\ P\left(A\right)=0.0737\end{array}$