Chapter 5.3, Problem 87E

(a)

To determine

Probability for the person contributed to the charity.

Answer to Problem 87E

Probability that the person contributed to the charity is 0.2240.

Explanation of Solution

Given information:

Figure showing probabilities for a charity by potential donors:

  

According to multiplication rule for independent events,

  $P\left(\text{A}\text{and}\text{B}\text{and}\text{C}\right)=P\left(\text{A}\cap \text{B}\cap \text{C}\right)=P\left(\text{A}\right)\times P\left(\text{B}\right)\times P\left(\text{C}\right)$

According to the addition rule for mutually exclusive events:

  $P\left(\text{A}\cup \text{B}\cup \text{C}\right)=P\left(\text{A}\text{or}\text{B}\text{or}\text{C}\right)=P\left(\text{A}\right)+P\left(\text{B}\right)+P\left(\text{C}\right)$

Let

R: Recent donor

P: Past donor

N: New prospect

PL: Pledge to contribute

C: Contributor

Now,

For recent donors:

Probability for recent donor,

  $P\left(\text{R}\right)=0.5$

Probability for recent donor pledge to contribute,

  $P\left(\text{PL}\right)=0.4$

Probability for recent donor makes contribution,

  $P\left(\text{C}\right)=0.8$

For past donors:

Probability for past donor,

  $P\left(\text{P}\right)=0.3$

Probability for past donor pledge to contribute,

  $P\left(\text{PL}\right)=0.3$

Probability for past donor makes contribution,

  $P\left(\text{C}\right)=0.6$

For new prospects:

Probability for new prospect,

  $P\left(\text{N}\right)=0.2$

Probability for new prospect pledge to contribute,

  $P\left(\text{PL}\right)=0.1$

Probability for new prospect makes contribution,

  $P\left(\text{C}\right)=0.5$

Now,

Since each person cannot become all three types of contributor,

Apply multiplication rule for each of the above three cases separately:

For recent donors:

  $P\left(\text{R}\cap \text{PL}\cap \text{C}\right)=P\left(\text{R}\right)\times P\left(\text{PL}\right)\times P\left(\text{C}\right)=0.5\times 0.4\times 0.8=0.1600$

For past donors:

  $P\left(\text{P}\cap \text{PL}\cap \text{C}\right)=P\left(\text{P}\right)\times P\left(\text{PL}\right)\times P\left(\text{C}\right)=0.3\times 0.3\times 0.6=0.0540$

For new prospect:

  $P\left(\text{N}\cap \text{PL}\cap \text{C}\right)=P\left(\text{N}\right)\times P\left(\text{PL}\right)\times P\left(\text{C}\right)=0.2\times 0.1\times 0.5=0.0100$

To get the probability for contributor,

Apply the addition rule for mutually exclusive events:

  $\begin{array}{l}P\left(\text{C}\right)=P\left(\text{R}\cap \text{PL}\cap \text{C}\right)+P\left(\text{P}\cap \text{PL}\cap \text{C}\right)+P\left(\text{N}\cap \text{PL}\cap \text{C}\right)\\ =0.1600+0.0540+0.0100\\ =0.2240\end{array}$

Thus,

The probability for the randomly selected person contributes to the charity is 0.2240.

(b)

To determine

Probability for the person contributed is a recent donor.

Answer to Problem 87E

Probability that the person contributed is a recent donor is approx. 0.7143.

Explanation of Solution

Given information:

Figure showing probabilities for a charity by potential donors:

  

Conditional probability definition:

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

From Part (a),

We have

Probability for the person contributes to charity,

  $P\left(\text{C}\right)=0.2240$

Probability for recent donor and pledged to contribute and contributor,

  $P\left(\text{R}\cap \text{PL}\cap \text{C}\right)=0.1600$

It is understood that

Recent donor who contributed was pledged to contribute.

Thus,

  $P\left(\text{R}\cap \text{PL}\cap \text{C}\right)=P\left(\text{R}\cap \text{C}\right)=0.1600$

Apply the conditional probability:

  $P\left(\text{R}|\text{C}\right)=\frac{P\left(\text{R}\cap \text{C}\right)}{P\left(\text{C}\right)}=\frac{0.1600}{0.2240}\approx 0.7143$

Thus,

The conditional probability for the person contributed is a recent donor is approx. 0.7143.