To solve the problem, we need to find the probability \( P(F|E) \), where:
- Event \( E \): The son is on one end of the lineup.
- Event \( F \): The father is in the middle of the lineup.
### Step 1: Determine the Sample Space
The family consists of three members: Mother (M), Father (F), and Son (S). The total number of arrangements (sample space) for these three members is given by the factorial of the number of members, which is \( 3! = 6 \).
The possible arrangements are:
1. M, F, S
2. M, S, F
3. F, M, S
4. F, S, M
5. S, M, F
6. S, F, M
So, the sample space \( S \) is:
\[ S = \{ (M, F, S), (M, S, F), (F, M, S), (F, S, M), (S, M, F), (S, F, M) \} \]
### Step 2: Identify Event \( E \)
Event \( E \) occurs when the son is on one end. The possible arrangements that satisfy this condition are:
1. S, F, M
2. S, M, F
3. M, F, S
4. F, M, S
Thus, the outcomes for event \( E \) are:
\[ E = \{ (S, F, M), (S, M, F), (M, F, S), (F, M, S) \} \]
The number of favorable outcomes for event \( E \) is 4.
### Step 3: Calculate \( P(E) \)
The probability of event \( E \) can be calculated as:
\[
P(E) = \frac{\text{Number of favorable outcomes for } E}{\text{Total outcomes in sample space}} = \frac{4}{6} = \frac{2}{3}
\]
### Step 4: Identify Event \( F \)
Event \( F \) occurs when the father is in the middle. The possible arrangements that satisfy this condition are:
1. M, F, S
2. S, F, M
Thus, the outcomes for event \( F \) are:
\[ F = \{ (M, F, S), (S, F, M) \} \]
The number of favorable outcomes for event \( F \) is 2.
### Step 5: Identify \( F \cap E \) (Intersection of \( F \) and \( E \))
Now, we need to find the intersection of events \( F \) and \( E \), which consists of outcomes that are common to both events:
- The arrangements that have the father in the middle and the son on one end are:
1. S, F, M
Thus, the intersection \( F \cap E \) is:
\[ F \cap E = \{ (S, F, M) \} \]
The number of favorable outcomes for \( F \cap E \) is 1.
### Step 6: Calculate \( P(F \cap E) \)
The probability of the intersection \( F \cap E \) can be calculated as:
\[
P(F \cap E) = \frac{\text{Number of favorable outcomes for } F \cap E}{\text{Total outcomes in sample space}} = \frac{1}{6}
\]
### Step 7: Calculate \( P(F|E) \)
Using the formula for conditional probability:
\[
P(F|E) = \frac{P(F \cap E)}{P(E)}
\]
Substituting the values we calculated:
\[
P(F|E) = \frac{\frac{1}{6}}{\frac{2}{3}} = \frac{1}{6} \times \frac{3}{2} = \frac{1}{4}
\]
### Final Answer
Thus, the probability \( P(F|E) \) is \( \frac{1}{4} \).
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