An unbiased six-sided die is thrown once. What is the probability of getting a number that is a multiple of 2 or a multiple of 5?

Introduction / Context:
This question asks for the probability that the outcome of a single roll of a fair six-sided die is either a multiple of 2 or a multiple of 5. It is a straightforward application of set union in a finite sample space and tests understanding of multiples and basic counting.

Given Data / Assumptions:

• The die is fair and has faces numbered 1, 2, 3, 4, 5, 6.
• Each face is equally likely, so each outcome has probability 1/6.
• We are interested in numbers that are multiples of 2 or multiples of 5.

Concept / Approach:
We list the outcomes that are multiples of 2, the outcomes that are multiples of 5, and then take their union. If there is any overlap, we make sure not to double count. Finally, the probability is the count of favourable outcomes divided by 6.

Step-by-Step Solution:
Possible outcomes on a six-sided die: 1, 2, 3, 4, 5, 6. Multiples of 2 in this set: 2, 4, 6. Multiples of 5 in this set: 5. Union of {multiples of 2} and {multiples of 5} = {2, 4, 5, 6}. There is no overlap between these sets because no number from 1 to 6 is simultaneously a multiple of both 2 and 5. Number of favourable outcomes = 4 (namely 2, 4, 5, 6). Total number of outcomes = 6. Required probability = favourable / total = 4 / 6. Simplify 4 / 6 by dividing numerator and denominator by 2 to get 2 / 3.

Verification / Alternative check:
We can check quickly by noticing that the only numbers not in the favourable set are 1 and 3, which are 2 of the 6 possible outcomes. Therefore, the complement probability is 2/6 = 1/3. The required probability is then 1 minus this complement: 1 - 1/3 = 2/3. This matches the earlier direct computation.

Why Other Options Are Wrong:
1/3: This is the probability of the complement event (getting 1 or 3) rather than the event of interest.
1/2: This would imply exactly three favourable outcomes, but there are four.
1/6: This would correspond to only one favourable outcome, which is clearly incorrect.
None of these: This is incorrect because 2/3 is explicitly listed and is the correct answer.

Common Pitfalls:
One common confusion is failing to work with the union of the sets correctly, leading to either missing a favourable outcome or double counting. Another mistake is misidentifying multiples, especially when quickly scanning numbers. Listing the possible outcomes and marking those that satisfy each condition helps avoid such errors.

Final Answer:
Therefore, the probability of getting a multiple of 2 or a multiple of 5 is 2/3.