Question

Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).

Hint:

Conditional probability is calculated using the formula \(P(A/B) = \frac{P(A \cap B)}{P(B)}\). It represents the probability of event \(A\) occurring given that event \(B\) has already occurred.

Answer 1

Step 1: Total number of arrangements
The total number of ways to arrange 3 people (Mother, Father, and Son) is given by: 
Total arrangements = 3! = 6.

Step 2: Event \(F\) (Father in the middle)
In this case, the Father is fixed in the middle position, and the other two people (Mother and Son) can be arranged in the remaining two positions. 
Number of arrangements for \(F\) = 2! = 2.

Step 3: Event \(E\) (Son on one end) given \(F\) (Father in the middle)
When the Father is in the middle, we need to arrange the Son and Mother on the two ends. There are 2 possible positions for the Son (either left or right end), and once the Son is placed, the Mother will occupy the remaining position. 
So, the number of favorable outcomes for \(E\) given \(F\) is 2 (Son on one end).

Step 4: Conditional Probability
The conditional probability \(P(E/F)\) is given by the ratio of the number of favorable outcomes for both \(E\) and \(F\) to the number of outcomes for \(F\): 
\[ P(E/F) = \frac{\text{Number of favorable outcomes for both \(E\) and \(F\)}}{\text{Number of outcomes for \(F\)}} = \frac{2}{2} = 1. \]

Final Answer:
\(P(E/F) = 1\)