A die is rolled. If the outcome is an odd number, then the probability that the number is prime, is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{2}{3} \)
This is a conditional probability. Let E = the number is prime = {2, 3, 5} and F = the number is odd = {1, 3, 5}.
\( \displaystyle E\cap F=\{3,5\} \)
\( \displaystyle P(E|F)=\frac{P(E\cap F)}{P(F)} \)
\( \displaystyle =\frac{2/6}{3/6}=\frac{2}{3} \)
Shortcut: Given that the outcome is odd, the reduced sample space is {1, 3, 5}; two of these three are prime, so the probability is 2/3.
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{2} \) — Incorrect. 1/2 is P(E) = 3/6, the probability of a prime without the condition.
Option (2) \( \displaystyle\frac{1}{3} \) — Incorrect. 1/3 is P(E ∩ F) = 2/6, which is not divided by P(F).
Option (4) \( \displaystyle\frac{1}{6} \) — Incorrect. The probability is 2/3, not 1/6.
Tip: 1 is not a prime number; counting it would give 3/3 = 1 here.