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)`

Answer

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 total number of children
Each family has 2 children, and there are 2 families. Therefore, the total number of children is:
\[ \text{Total children} = 2 \text{ (from family 1)} + 2 \text{ (from family 2)} = 4 \]

### Step 2: Identify the possible outcomes
Each child can either be a boy (B) or a girl (G). The possible combinations of 4 children can be represented as:
- GGGG
- GGGB
- GGBG
- GBGG
- BGGG
- GBBG
- GBGB
- BGBG
- BBGG
- BGBB
- BBGB
- BBBG
- BBBB

### Step 3: Calculate the probability of all four children being girls
The probability of all four children being girls (GGGG) is:
\[ P(\text{All 4 girls}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 4: Calculate the probability of having at least two girls
To find the probability of having at least two girls, we can count the favorable outcomes:
1. All 4 girls: GGGG
2. 3 girls and 1 boy: GGGB, GGBG, GBGG, BGGG (4 outcomes)
3. 2 girls and 2 boys: GGBB, GBGB, BGBG, BBGG (6 outcomes)

The total number of outcomes with at least 2 girls is:
\[ 1 \text{ (4 girls)} + 4 \text{ (3 girls)} + 6 \text{ (2 girls)} = 11 \]

### Step 5: Calculate the total number of outcomes
The total number of outcomes for 4 children is:
\[ 2^4 = 16 \]

### Step 6: Calculate the probability of at least two girls
The probability of having at least two girls is:
\[ P(\text{At least 2 girls}) = \frac{11}{16} \]

### Step 7: Use the conditional probability formula
We need to find the conditional probability:
\[ P(\text{All 4 girls} | \text{At least 2 girls}) = \frac{P(\text{All 4 girls})}{P(\text{At least 2 girls})} \]
Substituting the values we calculated:
\[ P(\text{All 4 girls} | \text{At least 2 girls}) = \frac{\frac{1}{16}}{\frac{11}{16}} = \frac{1}{11} \]

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