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. 24 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 to which the person belongs. Finally, those who make a pledge either do or don’t actually make a contribution. Suppose we randomly select a person who is called by the charity.

a. What is the probability that the person contributed to the charity?

b. Given that the person contributed, find the probability that he or she is a recent donor

According to multiplication rule for independent events,

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

According to the addition rule for mutually exclusive events:

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

Let

R: Recent donor

P: Past donor

N: New prospect

PL: Pledge to contribute

C: Contributor

Now,

For recent donors:

Probability for recent donor,

$P\left(\mathrm{R}\right)=0.5$

Probability for recent donor pledge to contribute,

$P\left(\mathrm{PL}\right)=0.4$

Probability for recent donor makes contribution,

$P\left(C\right)=0.8$

For past donors:

Probability for past donor,

$P\left(\mathrm{P}\right)=0.3$

Probability for past donor makes contribution,

$P\left(C\right)=0.6$

For new prospects:

Probability for new prospect,

$P(\mathrm{N})=0.2$

Probability for new prospect pledge to contribute,

$P\left(\mathrm{PL}\right)=0.1$

Probability for new prospect makes contribution,

$P\left(C\right)=0.5$

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:

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

For past donors:

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

For new prospect:

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

To get the probability for contributor,

Apply the addition rule for mutually exclusive events:

$P\left(\mathrm{C}\right)=P(\mathrm{R}\cap \mathrm{PL}\cap \mathrm{C})+P(\mathrm{P}\cap \mathrm{PL}\cap \mathrm{C})+P(\mathrm{N}\cap \mathrm{PL}\cap \mathrm{C})$

$=0.1600+0.0540+0.0100$

$=0.2240$

Thus,

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

From Part (a),

We have

Probability for the person contributes to charity,

$P\left(C\right)=0.2240$

Probability for recent donor and pledged to contribute and contributor,

$P(\mathrm{R}\cap \mathrm{PL}\cap \mathrm{C})=0.1600$

It is understood that

Recent donor who contributed was pledged to contribute.

Thus,

$P(\mathrm{R}\cap \mathrm{PL}\cap \mathrm{C})=P(\mathrm{R}\cap \mathrm{C})=0.1600$

Apply the conditional probability:

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

Thus,

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