Assumes that each born child is equally likely to be a boy or a girl. If two families have two children each, if conditional probability that all children are girls given that at least two are girls is `k,` then `1/k=`

Answer

To solve the problem, we need to calculate the conditional probability that all children are girls given that at least two are girls. Let's denote the events as follows:

- Let \( A \) be the event that all children are girls.
- Let \( B \) be the event that at least two children are girls.

We want to find \( P(A | B) \), which is given by the formula:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

### Step 1: Calculate \( P(A) \)

Since each child can be either a boy or a girl with equal probability, the probability of having all four children (two from each family) as girls is:

\[ P(A) = P(\text{4 girls}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 2: Calculate \( P(B) \)

Next, we need to find the probability of having at least two girls among the four children. We can find this by calculating the probabilities of having 0 or 1 girl and subtracting from 1.

- **Probability of 0 girls (all boys)**:
\[ P(\text{0 girls}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

- **Probability of 1 girl** (and 3 boys):
\[ P(\text{1 girl}) = \binom{4}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^3 = 4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]

Now, we can calculate \( P(B) \):

\[ P(B) = 1 - P(\text{0 girls}) - P(\text{1 girl}) = 1 - \frac{1}{16} - \frac{4}{16} = 1 - \frac{5}{16} = \frac{11}{16} \]

### Step 3: Calculate \( P(A \cap B) \)

The event \( A \) (all girls) is a subset of \( B \) (at least two girls), so:

\[ P(A \cap B) = P(A) = \frac{1}{16} \]

### Step 4: Calculate \( P(A | B) \)

Now we can substitute our values into the conditional probability formula:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{16}}{\frac{11}{16}} = \frac{1}{11} \]

### Step 5: Find \( \frac{1}{k} \)

Since we have \( P(A | B) = \frac{1}{11} \), we can say that \( k = \frac{1}{11} \). Therefore,

\[ \frac{1}{k} = 11 \]

### Final Answer

Thus, the value of \( \frac{1}{k} \) is \( 11 \).

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