Question

A die is rolled once. If the die shows odd number, then the probability of getting other than \( 5 \) is

• A. \( \dfrac{1}{6} \)
• B. \( \dfrac{5}{6} \)
• C. \( \dfrac{1}{3} \)
• D. \( \dfrac{2}{3} \)
• E. \( \dfrac{1}{5} \)

Hint:

In conditional probability, always first reduce the sample space based on the given condition, then compute probability within that reduced space.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a conditional probability problem. The sample space is reduced by the given condition. The phrase "If the die shows an odd number" tells us we are no longer considering all six possible outcomes of a die roll.
Step 2: Key Formula or Approach:
Let A be the event of "getting other than 5" and B be the event of "the die shows an odd number". We want to find the conditional probability \(P(A|B)\).
The formula is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).
Alternatively, we can use the reduced sample space method. The new sample space is the set of outcomes where the die shows an odd number. The favorable outcomes are the ones in this new sample space that are also "other than 5".
Step 3: Detailed Explanation:
Let S be the original sample space for rolling a die: \(S = \{1, 2, 3, 4, 5, 6\}\).
The given condition is that "the die shows an odd number". Let's call this event B.
The outcomes for event B form our new, reduced sample space, S'.
\[ S' = \{1, 3, 5\} \] The number of outcomes in the reduced sample space is \(n(S') = 3\).
Now, we need to find the probability of the event "getting other than 5" within this new sample space. Let's call this event A.
The favorable outcomes are the numbers in S' that are not 5.
Favorable outcomes = \(\{1, 3\}\).
The number of favorable outcomes is 2.
The required probability is the ratio of the number of favorable outcomes to the total number of outcomes in the reduced sample space.
\[ P(\text{other than 5 | odd}) = \frac{\text{Number of odd numbers that are not 5}}{\text{Total number of odd numbers}} \] \[ P = \frac{2}{3} \] Step 4: Final Answer:
The probability of getting other than 5, given that the die shows an odd number, is \(\frac{2}{3}\). This corresponds to option (D).