Q.31 The six faces of a cube (die) are numbered as 1, 2, 3, 4, 5 and 6, and it is rolled once.
An outcome is the observed number on the top face. If the probability of getting an
odd number as an outcome is twice that of an even number, then the probability of
getting a number less than 3 is ________.
(A) 1/9
(B) 2/9
(C) 1/3
(D) 4/9
Standard dice have equal probabilities, but this problem modifies outcomes where odd numbers appear twice as likely as even ones.
Solving reveals the exact probability for numbers less than 3.
Correct answer: (C) 1/3.
Problem Breakdown
A cube die shows numbers 1 through 6 on its faces and is rolled once.
Odd numbers (1, 3, 5) have probability twice that of even numbers (2, 4, 6).
Let the total probability of all even outcomes be x.
Then the total probability of all odd outcomes is 2x.
Since total probability equals 1:
3x = 1 ⇒ x = 1/3
There are three even faces and three odd faces, so:
- Probability of each even face = (1/3) ÷ 3 = 1/9
- Probability of each odd face = (2/3) ÷ 3 = 2/9
Calculating Probability of Numbers Less Than 3
Numbers less than 3 are:
- 1 (odd)
- 2 (even)
Therefore:
P(less than 3) = P(1) + P(2) = 2/9 + 1/9 = 3/9 = 1/3
This matches option (C).
Option Analysis
- (A) 1/9: Probability of a single even outcome like 2 alone.
- (B) 2/9: Probability of a single odd outcome like 1 alone.
- (C) 1/3: Correct combined probability of 1 and 2.
- (D) 4/9: Overestimates by including extra outcomes.
Standard vs Modified Die
In a fair die, each face has probability 1/6.
So, the probability of getting a number less than 3 is:
2/6 = 1/3
Interestingly, even with biased probabilities favoring odd numbers,
the result remains the same due to balanced contributions from 1 and 2.
