An unbiased cubic die with faces numbered 1 to 6 is thrown once. What is the probability of getting a number that is a multiple of 3 or a multiple of 4?

Introduction / Context: This question tests understanding of probability for outcomes on a single fair die. The event is defined using divisibility conditions. We must identify which face values are multiples of 3 or multiples of 4, then express the count of those faces as a fraction of the total possible outcomes.

Given Data / Assumptions:

• The die is fair and has six faces numbered 1, 2, 3, 4, 5 and 6.
• All six outcomes are equally likely.
• We require a result that is a multiple of 3 or a multiple of 4.

Concept / Approach: We use the basic probability formula: probability equals number of favourable outcomes divided by total outcomes. The sample space has 6 outcomes. The favourable set is the union of multiples of 3 and multiples of 4 among the numbers 1 to 6. Since the set is small, we can list and check each number directly, and then compute the ratio.

Step-by-Step Solution: Possible results when the die is thrown: 1, 2, 3, 4, 5, 6. Multiples of 3 in this set: 3 and 6. Multiples of 4 in this set: 4. Union of these values: 3, 4 and 6. Number of favourable outcomes = 3. Total number of possible outcomes = 6. Required probability = favourable / total = 3 / 6 = 1 / 2.

Verification / Alternative check: We can check quickly by noting that half of the numbers from 1 to 6 satisfy the required condition. There are three favourable numbers out of six. Because the die is fair, each number appears with equal chance, so a probability of 1 / 2 makes sense intuitively and matches our computed ratio of 3 / 6.

Why Other Options Are Wrong: 1/12 and 1/9 are too small and would suggest far fewer favourable outcomes than actually exist. 3/4 would require 4 or 5 favourable outcomes, but we only have 3 values that meet the divisibility rule.

Common Pitfalls: Errors usually occur when candidates incorrectly list the multiples, for example by including 2 or 5, or by thinking that 1 is a multiple of every integer. Another mistake is to count 12 as a possible outcome, which is not on a standard six faced die. Careful listing of all face values and checking each against the divisibility conditions prevents these mistakes.

Final Answer: The probability of getting a number that is a multiple of 3 or 4 is 1/2.