Question

A die was thrown twice and the sum of the numbers which appeared was found to be 6. Find the conditional probability that the number 4 appears at least once.

Hint:

In conditional probability, the sample space is reduced to the event that is known to have occurred. Instead of considering all 36 possible outcomes of two dice rolls, you only need to consider the 5 outcomes where the sum is 6. This simplifies the problem significantly.

Answer 1

Step 1: Understanding the Concept:
This problem asks for a conditional probability. We are given an event that has already occurred (the sum of the numbers is 6), and we need to find the probability of another event (the number 4 appears at least once) given this condition.
Step 2: Key Formula or Approach:
The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where:
- \(A\) is the event that the number 4 appears at least once.
- \(B\) is the event that the sum of the numbers is 6.
In terms of outcomes, this can be calculated as \( P(A|B) = \frac{\text{Number of outcomes in } A \cap B}{\text{Number of outcomes in } B} \).
Step 3: Detailed Explanation:
First, let's define the sample space for the given condition (Event B).
Event B: The sum of the numbers is 6.
The possible outcomes for two dice throws that sum to 6 are: \[ B = \{(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)\} \] The total number of outcomes for event B is \( n(B) = 5 \).
Next, let's define Event A.
Event A: The number 4 appears at least once.
Now, we need to find the intersection of A and B (\( A \cap B \)), which means we look for outcomes in B where the number 4 appears at least once.
From the set of outcomes for B, the ones that contain at least one 4 are: \[ A \cap B = \{(2, 4), (4, 2)\} \] The number of outcomes in \( A \cap B \) is \( n(A \cap B) = 2 \).
Step 4: Final Answer:
Using the formula for conditional probability: \[ P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{2}{5} \] So, the conditional probability that the number 4 appears at least once, given that the sum is 6, is \( \frac{2}{5} \).