A life insurance company insured 25,000 young boys, 14,000 young girls and 16,000 young adults. The probability of death within 10 years of a young boy, young girl and a young adult are 0.02, 0.03 and 0.15 respectively. One of the insured persons die. What is the probability that the dead person is a young boy?

Options

A. `36/165`
B. `25/166`
C. `26/165`
D. `32/165`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the number of young boys who may die within 10 years.
The probability of death for a young boy is given as 0.02, and the total number of young boys insured is 25,000.

\[ \text{Number of young boys who may die} = \text{Probability of death} \times \text{Total number of young boys} \]
\[ = 0.02 \times 25000 = 500 \]

### Step 2: Calculate the number of young girls who may die within 10 years.
The probability of death for a young girl is given as 0.03, and the total number of young girls insured is 14,000.

\[ \text{Number of young girls who may die} = \text{Probability of death} \times \text{Total number of young girls} \]
\[ = 0.03 \times 14000 = 420 \]

### Step 3: Calculate the number of young adults who may die within 10 years.
The probability of death for a young adult is given as 0.15, and the total number of young adults insured is 16,000.

\[ \text{Number of young adults who may die} = \text{Probability of death} \times \text{Total number of young adults} \]
\[ = 0.15 \times 16000 = 2400 \]

### Step 4: Calculate the total number of insured persons who may die.
Now, we will add the number of young boys, young girls, and young adults who may die.

\[ \text{Total number of deaths} = \text{Number of young boys who may die} + \text{Number of young girls who may die} + \text{Number of young adults who may die} \]
\[ = 500 + 420 + 2400 = 3320 \]

### Step 5: Calculate the probability that the dead person is a young boy.
The probability that the dead person is a young boy can be calculated using the formula:

\[ P(\text{young boy}) = \frac{\text{Number of young boys who may die}}{\text{Total number of deaths}} \]
\[ = \frac{500}{3320} \]

### Step 6: Simplify the probability.
Now we will simplify the fraction:

\[ = \frac{500}{3320} = \frac{25}{166} \]

Thus, the probability that the dead person is a young boy is \(\frac{25}{166}\).

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