What is the probability that in a family of 4 children there will be at least one boy?

Options

A. `(15)/(16)`
B. `(3)/(8)`
C. `(1)/(16)`
D. `(7)/(8)`

Answer

To solve the problem of finding the probability that in a family of 4 children there will be at least one boy, we can follow these steps:

### Step 1: Understand the total possibilities
Each child can either be a boy (B) or a girl (G). Therefore, for 4 children, the total number of combinations of boys and girls can be calculated using the formula \(2^n\), where \(n\) is the number of children.

**Calculation:**
\[ \text{Total possibilities} = 2^4 = 16 \]

### Step 2: Identify the complementary event
To find the probability of having at least one boy, it is easier to first calculate the probability of the complementary event, which is having no boys at all (i.e., all girls).

### Step 3: Calculate the probability of all girls
If all 4 children are girls, there is only one combination: GGGG. Thus, the probability of having all girls can be calculated as follows:

**Calculation:**
\[ \text{Probability of all girls} = \frac{\text{Number of ways to have all girls}}{\text{Total possibilities}} = \frac{1}{16} \]

### Step 4: Calculate the probability of at least one boy
Now, we can find the probability of having at least one boy by subtracting the probability of having all girls from 1.

**Calculation:**
\[ \text{Probability of at least one boy} = 1 - \text{Probability of all girls} = 1 - \frac{1}{16} = \frac{15}{16} \]

### Step 5: Conclusion
Thus, the probability that in a family of 4 children there will be at least one boy is:

\[ \frac{15}{16} \]

### Final Answer
The correct answer is \( \frac{15}{16} \).

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