Out of 10,000 families with 4 children each , the probability number of families all of whose children are daughters is

Options

A. 625
B. 1250
C. 2500
D. 9375

Answer

To solve the problem of finding the expected number of families with 4 daughters out of 10,000 families, we can follow these steps:

### Step 1: Determine the probability of having a daughter
Since each child can either be a boy or a girl, the probability of having a daughter (P(girl)) is:
\[ P(\text{girl}) = \frac{1}{2} \]

### Step 2: Calculate the probability of having 4 daughters in a family
To find the probability of a family having all 4 children as daughters, we multiply the probability of having a daughter for each of the 4 children:
\[ P(\text{4 daughters}) = P(\text{girl}) \times P(\text{girl}) \times P(\text{girl}) \times P(\text{girl}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 3: Calculate the expected number of families with 4 daughters
The expected number of families with 4 daughters can be calculated using the formula:
\[ E(X) = N \times P \]
where \(N\) is the total number of families (10,000) and \(P\) is the probability of having 4 daughters (\(\frac{1}{16}\)):
\[ E(X) = 10,000 \times \frac{1}{16} = 625 \]

### Final Answer
The expected number of families with all daughters is:
\[ \boxed{625} \]