Chapter 19, Problem 19.42TY

a.

To determine

To find: The probability that the randomly selected victim was male.

Answer to Problem 19.42TY

The probability that the randomly selected victim was male is 0.8142.

Explanation of Solution

Given info:

The table shows results, in which death of 1,818 females was due to accidents, death of 457 females was due to homicide and death of 345 females was due to suicide. Also, death of 6,457 males was due to accidents, death of 2,870 males was due to homicide and death of 2,152 males was due to suicide.

Calculation:

The total number of people is obtained as shown in Table (1).

	Female	Male	Total
Accidents	1,818	6,457	8,275
Homicide	457	2,870	3,327
Suicide	345	2,152	2,497
Total	2,620	11,479	14,099

Table (1)

Let event A getting the victim be male.

The formula for probability of event A is as follows,

$P\left(A\right)=\frac{\text{Number of ways event }A\text{ occurs}}{\text{Sample size}}$

Substitute 11,479 for ‘Number of ways of event A occurs’ and 14,099 for ‘Sample size’

$\begin{array}{c}P\left(A\right)=\frac{11,479}{14,099}\\ =0.8142\end{array}$

Thus, the probability that the randomly selected victim was male is 0.8142.

b.

To determine

To find: The probability that the victim was male, given that the death was accidental.

Answer to Problem 19.42TY

The probability that the victim was male, given that the death was accidental is 0.7804.

Explanation of Solution

Calculation:

Let event B denote the death was accidental and the event A denote the victim was male.

Probability:

The formula for probability of event B is as follows,

$P\left(B\right)=\frac{\text{Number of ways event }B\text{ occurs}}{\text{Sample size}}$

Substitute 8,275 for ‘Number of ways of event B occurs’ and 14,099 for ‘Sample size’

$\begin{array}{c}P\left(B\right)=\frac{8,275}{14,099}\\ =0.5869\end{array}$

The formula for probability of event A and B is as follows,

$P\left(A\text{ and }B\right)=\frac{\text{Number of ways event }A\text{ and }B\text{ occurs}}{\text{Number of simple events}}$

Substitute 6,457 for ‘Number of ways of event A and B occurs’ and 14,099 for ‘Number of simple events’

$\begin{array}{c}P\left(A\text{ and }B\right)=\frac{6,457}{14,099}\\ =0.4580\end{array}$

The formula for conditional probability is as follows,

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

Substitute 0.5869 for $P\left(B\right)$ and 0.4580 for $P\left(A\text{ and }B\right)$,

$\begin{array}{c}P\left(A|B\right)=\frac{0.4580}{0.5869}\\ =0.7804\end{array}$

Therefore, the probability that the victim was male, given that the death was accidental is 0.7804.

Interpretation:

There is 0.7804 probability that the victim was male, given that the death was accidental.

c.

To determine

To explain: Whether the sex and type of death are independent or not.

Answer to Problem 19.42TY

No, the sex and type of death are not independent.

Explanation of Solution

Calculation:

From, part (a), it is observed that the probability that the victim was male, which is $P\left(A\right)$ 0.8142 and from part (b), it is observed that $P\left(B\right)$ is 0.5869 and $P\left(A\text{ and }B\right)$ is 0.4580.

Independent Events:

If events A and B are independent, then the probability of occurring of event B is not affected by event A, that is $P\left(A\text{ and }B\right)$ becomes $P\left(A\right)\times P\left(B\right)$.

$\begin{array}{c}P\left(A\right)\times P\left(B\right)=0.8142\times 0.5869\\ =0.4779\\ P\left(A\text{ and }B\right)=0.\text{458}0\end{array}$

$\begin{array}{c}0.4580\ne 0.4779\\ P\left(A\text{ and }B\right)\ne P\left(A\right)\times P\left(B\right)\end{array}$

Thus, events A and B are not independent of each other, that is the sex and type of death is not independent.