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According to multiplication rule for independent events,

$P(\mathrm{A}\text{and}\mathrm{B}\text{and}\mathrm{C})=P(\mathrm{A}鈭�\mathrm{B}鈭�\mathrm{C})=P(\mathrm{A})脳P(\mathrm{B})脳P\left(\mathrm{C}\right)$

According to the addition rule for mutually exclusive events:

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

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:

$P(\mathrm{R}鈭�\mathrm{PL}鈭�\mathrm{C})=P\left(\mathrm{R}\right)脳P\left(\mathrm{PL}\right)脳P\left(\mathrm{C}\right)=0.5脳0.4脳0.8=0.1600$

For past donors:

$P(\mathrm{P}鈭�\mathrm{PL}鈭�\mathrm{C})=P\left(\mathrm{P}\right)脳P\left(\mathrm{PL}\right)脳P\left(\mathrm{C}\right)=0.3脳0.3脳0.6=0.0540$

For new prospect:

$P(\mathrm{N}鈭�\mathrm{PL}鈭�\mathrm{C})=P(\mathrm{N})脳P\left(\mathrm{PL}\right)脳P\left(\mathrm{C}\right)=0.2脳0.1脳0.5=0.0100$

To get the probability for contributor,

Apply the addition rule for mutually exclusive events:

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

$=0.1600+0.0540+0.0100$

$=0.2240$

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,

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

It is understood that

Recent donor who contributed was pledged to contribute.

Thus,

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

Apply the conditional probability:

$P(\mathrm{R}鈭�\mathrm{C})=\frac{P(\mathrm{R}鈭�\mathrm{C})}{P\left(\mathrm{C}\right)}=\frac{0.1600}{0.2240}鈮�0.7143$

Thus,

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