Chapter 4, Problem 10RE

a.

To determine

Compute the probability that selected patient is a woman.

Answer to Problem 10RE

The probability that selected patient is a woman is 0.4035.

Explanation of Solution

Calculation:

It is given that 644,404 patients were hospitalized for heart attack. The table provides idea about number of male and female patients hospitalized in each age group.

The probability of the event A can be obtained by the formula:

$P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Number of outcomes in}\text{sample space}}$

Here, event A denotes that selected patient is a woman. Among the 644,404 patients, 260,005 of them are woman.

Substitute 260,005 for “number of outcomes in A” and 644,404 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(A\right)=\frac{260,005}{644,404}\\ =0.40348\\ =0.4035\end{array}$

Thus, the probability that selected patient is a woman is 0.4035.

b.

To determine

Compute the probability that selected patient is aged 45-64.

Answer to Problem 10RE

The probability that selected patient is aged 45-64 is 0.3647.

Explanation of Solution

Calculation:

Event B denotes that selected patient is aged 45-64. Among the 644,404 patients,  235,006 of them are woman.

Substitute 235,006 for “number of outcomes in B” and 644,404 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(B\right)=\frac{235,006}{644,404}\\ =0.36468\\ =0.3647\end{array}$

Thus, the probability that selected patient is aged 45-64 is 0.3647.

c.

To determine

Compute the probability that selected patient is a woman and aged 45-64.

Answer to Problem 10RE

The probability that selected patient is a woman and aged 45-64 is 0.1066.

Explanation of Solution

Calculation:

Event A denotes that selected patient is a woman and event B denotes that selected patient is aged 45-64. Event ‘A and B’ denote that selected patient is a woman and aged 45-64.

Among the 644,404 patients, 68,666 of them are woman and aged 45-64.

Substitute 68,666 for “number of outcomes in ‘A and B’” and 644,404 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(A\text{and }B\right)=\frac{68,666}{644,404}\\ =0.10655\\ =0.1066\end{array}$

Thus, the probability that selected patient is a woman and aged 45-64 is 0.1066.

d.

To determine

Compute the probability that selected patient is a woman or aged 45-64.

Answer to Problem 10RE

The probability that selected patient is a woman or aged 45-64 is 0.6616.

Explanation of Solution

Calculation:

Event A denotes that selected patient is a woman and event B denotes that selected patient is aged 45-64. Event ‘A and B’ denote that selected patient is a woman and aged 45-64.

The probability that selected patient is a woman or aged 45-64 can be expressed as, $P\left(A\text{or }B\right)$.

General Addition rule:

For any two events A and B the general addition rule states that

$P\left(A\text{or }B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and }B\right)$

From parts (a), (b), and (c), $P\left(A\right)=0.4035\text{, }P\left(B\right)=0.3647,\text{and}P\left(A\text{and }B\right)=0.1066$.

Substitute these values in the general addition rule.

Therefore,

$\begin{array}{c}P\left(A\text{or }B\right)=0.4035+0.3647-0.1066\\ =0.6616\end{array}$

Thus, the probability that selected patient is a woman or aged 45-64 is 0.6616.

e.

To determine

Compute the probability that selected patient is a woman given that the patient is aged 45-64.

Answer to Problem 10RE

The probability that selected patient is a woman given that the patient is aged 45-64 is 0.2922.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

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

From part (b), $P\left(B\right)=0.3647$.

From part (c), $P\left(A\text{and }B\right)=0.1066$.

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(A|B\right)=\frac{0.1066}{0.3647}\\ =0.2922\end{array}$

Thus, the probability that selected patient is a woman given that the patient is aged 45-64 is 0.2922.

f.

To determine

Compute the probability that selected patient is aged 45-64 given that the patient is a woman.

Answer to Problem 10RE

The probability that selected patient is aged 45-64 given that the patient is a woman is 0.2641.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

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

From part (a), $P\left(A\right)=0.4035$.

From part (c), $P\left(A\text{and }B\right)=0.1066$.

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(B|A\right)=\frac{0.1066}{0.4035}\\ =0.2641\end{array}$

Thus, the probability that selected patient is aged 45-64 given that the patient is a woman is 0.2641.