An unbiased six faced die is tossed once. What is the probability that the number appearing on the upper face is a multiple of 3?

Introduction / Context:
This question is a simple probability problem involving the outcomes of rolling a single fair die. We are asked for the probability that the outcome is a multiple of 3. Problems like this provide practice in identifying favorable outcomes in a small finite sample space and applying the basic probability formula for equally likely outcomes.

Given Data / Assumptions:

• The die is fair and has six faces.
• The faces are numbered 1, 2, 3, 4, 5 and 6.
• All six outcomes are equally likely.
• We are looking for outcomes that are multiples of 3.

Concept / Approach:
We determine which outcomes from the set {1, 2, 3, 4, 5, 6} are multiples of 3. The probability of the event is then the count of these favorable outcomes divided by the total number of possible outcomes, which is 6. This is a direct application of the classical probability definition for equally likely outcomes.

Step-by-Step Solution:
Step 1: List all possible outcomes from rolling the die: 1, 2, 3, 4, 5 and 6.Step 2: Identify which of these numbers are multiples of 3. Multiples of 3 in this range are 3 and 6.Step 3: Count the favorable outcomes. There are 2 favorable outcomes, namely 3 and 6.Step 4: Count the total number of outcomes, which is 6.Step 5: Probability of getting a multiple of 3 is favorable outcomes divided by total outcomes = 2 / 6.Step 6: Simplify 2 / 6 by dividing numerator and denominator by 2 to obtain 1 / 3.
Verification / Alternative check:
As a quick check, we can also look at the complement event, which is the event that the outcome is not a multiple of 3. Non multiples of 3 in this set are 1, 2, 4 and 5, which are 4 out of 6 outcomes, giving probability 4 / 6 = 2 / 3. Adding the probabilities of multiples and non multiples of 3, that is 1 / 3 + 2 / 3, yields 1, confirming that our classification and probability calculations are complete and consistent.

Why Other Options Are Wrong:
1/2 would imply that half of the die faces are multiples of 3, which would require 3 favorable outcomes, but there are only 2. 3/4 and 3/2 exceed 1/2 and 1 respectively, and therefore cannot represent this event. 2/3 is the probability of obtaining a non multiple of 3, not the probability of obtaining a multiple of 3, so it is the complement of the desired probability.

Common Pitfalls:
Some students may mistakenly count 0 as an outcome or misidentify the multiples of 3. Others might try to overcomplicate the problem by thinking of divisibility patterns without simply listing the small set of outcomes. In basic dice problems, it is safest to explicitly write down the faces and count the ones that satisfy the required property.

Final Answer:
The probability that the number appearing on the die is a multiple of 3 is 1/3.