Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls , is

Options

A. `(1)/(17)`
B. `(1)/(12)`
C. `(1)/(10)`
D. `(1)/(11)`

Explanation

To solve the problem, we need to find the conditional probability that all children are girls given that at least two are girls. Let's break this down step by step.

### Step 1: Define the Sample Space
Each child can either be a boy (B) or a girl (G). For two families with two children each, the total number of children is 4. The possible combinations of children can be represented as follows:
- GGGG
- GGGB
- GGBG
- GBGG
- BGGG
- GBBG
- GBGB
- BBGG
- BGBG
- BGGG
- GBBB
- BBBB

### Step 2: Count the Total Outcomes
The total number of outcomes for 4 children (2 families with 2 children each) is \(2^4 = 16\).

### Step 3: Define the Event of Interest
Let:
- A = Event that all children are girls (i.e., GGGG)
- B = Event that at least two children are girls

### Step 4: Calculate the Probability of Event A
The probability of event A (all children are girls) is:
\[ P(A) = \frac{1}{16} \]
This is because there is only one way to have all four children as girls (GGGG) out of 16 total combinations.

### Step 5: Calculate the Probability of Event B
Now, we need to find the probability of event B (at least two children are girls). We can find this by counting the outcomes that have at least two girls:
- 2 girls: GGBB, GBGB, BGGG (6 combinations)
- 3 girls: GGBB, GBG, BGG (4 combinations)
- 4 girls: GGGG (1 combination)

Counting these, we have:
- 6 (for 2 girls) + 4 (for 3 girls) + 1 (for 4 girls) = 11 outcomes

Thus, the probability of event B is:
\[ P(B) = \frac{11}{16} \]

### Step 6: Calculate the Conditional Probability
Now we can use the formula for conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Since event A (all girls) is a subset of event B (at least two girls), we have:
\[ P(A \cap B) = P(A) = \frac{1}{16} \]

Now substituting into the conditional probability formula:
\[ P(A | B) = \frac{P(A)}{P(B)} = \frac{\frac{1}{16}}{\frac{11}{16}} = \frac{1}{11} \]

### Final Answer
Thus, the conditional probability that all children are girls given that at least two are girls is:
\[ \boxed{\frac{1}{11}} \]