If an unbiased six sided die is rolled once, what are the odds in favour of getting a number that is a multiple of 3?

Introduction / Context:
This problem uses the concept of odds in favour instead of simple probability. We roll a fair six sided die once and look at the event that the outcome is a multiple of 3. We must express the answer as odds in favour of that event.

Given Data / Assumptions:

• The die is fair, with faces numbered 1, 2, 3, 4, 5 and 6.
• All six outcomes are equally likely.
• The favourable event is that the number shown is a multiple of 3.
• Odds in favour are defined as (number of favourable outcomes) : (number of unfavourable outcomes).

Concept / Approach:
First, we identify the multiples of 3 among the outcomes. Then we count the favourable outcomes and the unfavourable outcomes. The odds in favour of an event are a ratio of counts, not a fraction of probabilities, although they can be related. Here we simply form that ratio in simplest integer terms.

Step-by-Step Solution:
Possible die outcomes: 1, 2, 3, 4, 5, 6.Multiples of 3 among these are 3 and 6.Number of favourable outcomes = 2 (namely 3 and 6).Number of unfavourable outcomes = 6 - 2 = 4 (namely 1, 2, 4 and 5).Odds in favour of the event = favourable : unfavourable = 2 : 4.Simplify the ratio 2 : 4 by dividing both sides by 2 to get 1 : 2.

Verification / Alternative check:
The probability of getting a multiple of 3 is 2/6 = 1/3. The complement probability is 4/6 = 2/3. Odds in favour can also be expressed as P(E) : P(not E) = (1/3) : (2/3) which simplifies to 1 : 2, identical to the ratio found by direct counting. This confirms the correctness of the odds.

Why Other Options Are Wrong:
The ratio 2:1 would mean there are twice as many favourable outcomes as unfavourable ones, which is not true here. The ratio 1:3 would correspond to 1 favourable and 3 unfavourable outcomes, and 3:1 would correspond to 3 favourable and 1 unfavourable. Neither matches the actual counts 2 and 4. Only 1:2 correctly represents the odds in favour based on the actual outcome counts.

Common Pitfalls:
Students often confuse odds in favour with probability. The probability of the event is 1/3, but odds in favour are 1:2. Writing 1/3 as the answer when odds are requested is not correct. It is important to distinguish the representation of odds, which always compares favourable and unfavourable counts, from probability, which compares favourable counts to the total number of outcomes.

Final Answer:
The odds in favour of getting a multiple of 3 are 1:2.