Question

A six-faced die is rolled twice. Then the probability that an even number turns up at the first throw, given that the sum of the throws is 8, is

• A. 5/36
• B. 3/36
• C. 3/5
• D. 2/5

Hint:

For conditional probability, start by listing all outcomes that satisfy the "given" condition. This is your new, smaller sample space. Then, count how many of these specific outcomes also satisfy the event you are interested in.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a conditional probability problem. We are asked to find the probability of an event A (even number on the first throw) happening, given that another event B (the sum of throws is 8) has already happened. The sample space is reduced to only the outcomes where the sum is 8.

Step 2: Key Formula or Approach:
The conditional probability of event A given event B is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\text{Number of outcomes in A and B}}{\text{Number of outcomes in B}} \]
Step 3: Detailed Explanation:
Let's define the events:

Event B (the given condition): The sum of the two throws is 8.
Event A: An even number turns up on the first throw. \end{itemize} First, we list all possible outcomes where the sum of the two throws is 8. This is our reduced sample space (event B). The possible pairs are (first throw, second throw): \[ B = \{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\} \] The total number of outcomes in event B is \(n(B) = 5\).
Next, we identify which of these outcomes also satisfy event A (an even number on the first throw). These are the outcomes in the intersection of A and B (\(A \cap B\)). Looking at the set B, the outcomes with an even number on the first throw are: \[ A \cap B = \{(2, 6), (4, 4), (6, 2)\} \] The number of outcomes in \(A \cap B\) is \(n(A \cap B) = 3\).
Now, we can calculate the conditional probability: \[ P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{3}{5} \]
Step 4: Final Answer:
The probability that an even number turns up at the first throw, given that the sum is 8, is \(\frac{3}{5}\).