Chapter 5.3, Problem 98E

a.

To determine

To find:the percentage of calls results in a contribution.

Answer to Problem 98E

  $22.40\%$

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{Recent}\right)=50\%=0.5\\ P\left(\text{Past}\right)=30\%=0.3\\ P\left(\text{New}\right)=20\%=0.2\end{array}$ .

Concept used:

Multiplication rule $P\left(A\cap B\cap C\right)=P\left(A\right)\times P\left(B\right)\times P\left(C\right)$

Calculation:

Multiply the corresponding probabilities given along the branches:

  $\begin{array}{l}P\left(\text{Recent}\cap \text{Pledge}\cap \text{Contribute}\right)=0.5\times 0.4\times 0.8=0.1600\\ P\left(\text{Past}\cap \text{Pledge}\cap \text{Contribute}\right)=0.3\times 0.3\times 0.6=0.054\\ P\left(\text{New}\cap \text{Pledge}\cap \text{Contribute}\right)=0.2\times 0.1\times 0.5=0.01\end{array}$

Add the corresponding probabilities:

  $\begin{array}{l}P\left(\text{Contribute}\right)\\ =0.1600+0.0540+0.2240\\ =0.2240=22.40\%\end{array}$

Hence,the percentage of calls results in a contribution is $22.40\%$ .

b.

To determine

To find: the percentage who contribute are recent donor.

Answer to Problem 98E

The percentage who contribute are recent donor is $71.43\%$ .

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{Recent}\right)=50\%=0.5\\ P\left(\text{Past}\right)=30\%=0.3\\ P\left(\text{New}\right)=20\%=0.2\end{array}$ .

Concept used:

Definition conditional probability:

  $P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}$

Calculation:

From part (a): $P\left(\text{Contribute}\right)=22.40\%$

Definition conditional probability:

  $P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}$

Thus obtained:

  $\begin{array}{l}P\left(\text{Recent}|\text{Contribute}\right)=\frac{P\left(\text{Recent}\cap \text{Contribute}\right)}{P\left(\text{Contribute}\right)}\\ =\frac{0.1600}{0.2240}\approx 0.7143=71.43\%\end{array}$

Hence,the percentage who contribute are recent donor is $71.43\%$ .