Tossing a coin and rolling a die are two separate events. What is the probability of obtaining a tail on the coin and an even number on the die?

Analysis

This is a problem involving independent events in probability. Two events are independent if the outcome of one does not affect the outcome of the other. Here, tossing a coin has no influence on rolling a die.

Concept

When two events, $A$ and $B$, are independent, the probability of both occurring (the intersection) is the product of their individual probabilities:

Step-by-Step Solution

Step 1: Find the probability of obtaining a tail on the coin ($P(A)$) A fair coin has two sides: Head (H) and Tail (T).

• Total outcomes = 2
• Favorable outcome (Tail) = 1
• $P(\text{Tail}) = \frac{1}{2}$

Step 2: Find the probability of obtaining an even number on the die ($P(B)$) A standard six-sided die has the numbers {1, 2, 3, 4, 5, 6}.

• Total outcomes = 6
• Even numbers = {2, 4, 6}
• Number of favorable outcomes = 3
• $P(\text{Even}) = \frac{3}{6} = \frac{1}{2}$

Step 3: Calculate the combined probability Since the events are independent, multiply the two probabilities:

Conclusion

The probability of obtaining a tail on the coin and an even number on the die is $\frac{1}{4}$.

Exam Tip: Whenever you see the word "and" in probability questions involving independent events, it is a signal to multiply the probabilities. If you see the word "or", you generally add them.