Sure! Let's find the probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring.
Step-by-Step Solution:
1. Determine the probability of each individual event:
- Event [tex]\( A \)[/tex]: The coin landing on heads.
Since a coin has two sides (heads and tails) and each side is equally likely to occur,
[tex]\[
P(A) = \frac{1}{2}
\][/tex]
- Event [tex]\( B \)[/tex]: The die landing on 1, 3, or 6.
A six-sided die has six possible outcomes: 1, 2, 3, 4, 5, and 6. The favorable outcomes 1, 3, and 6 are three out of these six possible outcomes.
Thus,
[tex]\[
P(B) = \frac{3}{6} = \frac{1}{2}
\][/tex]
2. Calculate the combined probability of both events occurring:
Since events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we use the formula for the probability of both events occurring together:
[tex]\[
P(A \text{ and } B) = P(A) \cdot P(B)
\][/tex]
3. Substitute the probabilities into the formula:
[tex]\[
P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}
\][/tex]
So, the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] will occur is:
[tex]\[
P(A \text{ and } B) = \frac{1}{4}
\][/tex]
Answer:
[tex]\[
P(A \text{ and } B) = \frac{1}{4}
\][/tex]
Thus, in simplest form, the probability [tex]\( P(A \text{ and } B) \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
