Sure, let's work through this step-by-step to determine the probability that both events will occur.
Step 1: Determine the probability of event A
Event A is the coin landing on heads. A fair coin has two possible outcomes: heads or tails. Therefore, the probability of landing on heads (event A) is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
Step 2: Determine the probability of event B
Event B is the die landing on either 1, 3, or 6. A fair six-sided die has six possible outcomes: 1, 2, 3, 4, 5, and 6. The favorable outcomes for event B (1, 3, or 6) are three in total. Thus, the probability of event B is:
[tex]\[ P(B) = \frac{3}{6} = \frac{1}{2} \][/tex]
Step 3: Determine the probability of both events occurring
Since events A and B are independent, the probability of both events occurring (A and B) is obtained by multiplying their individual probabilities:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the values we obtained:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \][/tex]
Therefore, the probability that both events will occur is:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
