Question

Fundraising by telephone Tree diagrams can organize problems having more than two stages. The figure shows probabilities for a charity calling potential

donors by telephone.$21$Each person called is either a recent donor, a past donor, or a new prospect. At the next stage, the person called either does or does not pledge to contribute, with conditional probabilities that depend on the donor class the person belongs to. Finally, those who make a pledge either do or don’t actually make a contribution.

(a) What percent of calls result in a contribution?

(b) What percent of those who contribute are recent donors?

Short answer

Part (a) $22.40\%$

Part (b) The percentage who contribute are recent donor is $71.43\%$

Step-by-step solution

Part (a) Step 1. Given Information

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

Part (a) Step 2. Concept used

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

Part (a) Step 3. Calculation

Multiply the probability along the branches with the following formula:

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

Fill in the blanks with the corresponding probabilities:

$P\left(Contribute\right)=0.1600+0.0540+0.2240=0.2240=22.40\%$

As a result, $22.40\%$ of phone calls result in a contribution.

Part (b) Step 1. Concept

Conditional probability:$P\left(A\right|B)=P(A\cap B)/P(A)$

Part (b) Step 2. Calculation

$P\left(Contribute\right)=22.40\%$

Conditional probability: $P\left(A\right|B)=P(A\cap B)/P(A)$

As a result, we have:

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

As a result, $71.43\%$ of those who contribute are new donors.