A coin is tossed and a fair six-sided die is rolled. What is the probability of obtaining a tail on the coin and a 4 on the die in the same trial?

Introduction / Context:
This problem combines two simple random experiments: tossing a coin and rolling a fair six-sided die. We are interested in the probability of a specific combined event, namely getting a tail on the coin and a 4 on the die at the same time. This tests understanding of independent events and how to multiply probabilities.

Given Data / Assumptions:

• The coin is fair and has two outcomes: Head (H) and Tail (T).
• The die is fair with six faces numbered 1 to 6.
• The coin toss and the die roll are independent events.
• We want the probability of the combined event: T on the coin and 4 on the die.

Concept / Approach:
When two events are independent, the probability of both occurring together is the product of their individual probabilities. Therefore, we find P(T) for the coin and P(4) for the die, then multiply them. Alternatively, we can think in terms of an overall sample space of 2 * 6 = 12 equally likely outcomes and count the favourable outcome directly.

Step-by-Step Solution:
The probability of getting a tail on the coin, P(T) = 1/2. The probability of getting a 4 on the die, P(4) = 1/6. Since the two experiments are independent, P(T and 4) = P(T) * P(4). Compute: P(T and 4) = (1/2) * (1/6) = 1/12. Hence, the required probability is 1/12.

Verification / Alternative check:
We can construct the sample space explicitly. All possible ordered pairs (coin outcome, die outcome) are: (H,1), (H,2), ..., (H,6), (T,1), (T,2), ..., (T,6). There are 12 equally likely outcomes in total. Only one of these is (T,4), which corresponds to a tail and a 4. Therefore, probability = 1 favourable outcome / 12 total outcomes = 1/12, confirming the earlier calculation.

Why Other Options Are Wrong:
1/2: This might be chosen by someone who only considers the coin and ignores the die, but it does not account for the combined event.
2/3 and 3/4: These values are very large and would mean that the event happens most of the time, which is clearly not reasonable for a specific combined outcome.
None of these: This is incorrect because 1/12 is one of the listed options and is correct.

Common Pitfalls:
A common mistake is to add probabilities instead of multiplying them for independent events. Another error is to forget that the die has six outcomes and to assume some incorrect denominator. Some learners also confuse the phrase "and" with "or" in probability and calculate the probability of tail or 4, which is a completely different event.

Final Answer:
Therefore, the probability of obtaining a tail on the coin and a 4 on the die in the same trial is 1/12.