In a hospital, on the same day-5 children were born to different couples. What is the probability of all 5 children being boys?

Options

A. `1//32`
B. `1//16`
C. `5//16`
D. `5//32`

Answer

To solve the problem of finding the probability of all 5 children being boys, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Probability of Each Child**:
Each child can either be a boy or a girl. The probability of having a boy (P(B)) is 0.5, and the probability of having a girl (P(G)) is also 0.5.

**Hint**: Remember that the probability of two mutually exclusive events (boy or girl) must add up to 1.

2. **Identify the Total Number of Children**:
In this scenario, there are 5 children born on the same day to different couples.

**Hint**: Count the total number of independent events (children) involved in the scenario.

3. **Determine the Independence of Events**:
The birth of each child is an independent event. This means that the outcome of one child's gender does not affect the others.

**Hint**: Independent events allow us to multiply the probabilities of each event occurring.

4. **Calculate the Probability of All Boys**:
Since the events are independent, the probability of all 5 children being boys is calculated by multiplying the probability of having a boy for each child:
\[ P(\text{All boys}) = P(B) \times P(B) \times P(B) \times P(B) \times P(B) = (0.5)^5 \]

**Hint**: When dealing with independent events, use multiplication for the total probability.

5. **Perform the Calculation**:
Now, calculate \( (0.5)^5 \):
\[ (0.5)^5 = \frac{1}{2^5} = \frac{1}{32} \]

**Hint**: Remember to convert the decimal to a fraction if needed for clarity.

6. **Final Probability**:
The final probability of all 5 children being boys is:
\[ P(\text{All boys}) = \frac{1}{32} \approx 0.03125 \]

**Hint**: Express your final answer in both fraction and decimal form for completeness.

### Summary:
The probability of all 5 children being boys is \( \frac{1}{32} \) or approximately 0.03125.