To find the probability that both events A and B will occur, we'll follow these steps:
1. Determine the probability of Event A: The first die does NOT land on 5.
Since a six-sided die has 6 outcomes (1, 2, 3, 4, 5, 6) and only one of these outcomes is a 5, there are 5 favorable outcomes where the first die does not land on 5 (i.e., 1, 2, 3, 4, 6).
Hence, the probability of Event A occurring is:
[tex]\[
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{6}
\][/tex]
2. Determine the probability of Event B: The second die lands on 4.
Similarly, since a six-sided die has 6 outcomes, and only one of these outcomes is a 4, there is exactly 1 favorable outcome.
Thus, the probability of Event B occurring is:
[tex]\[
P(B) = \frac{1}{6}
\][/tex]
3. Find the probability that both events A and B will occur.
Since the events are independent, the probability that both events occur is the product of the probabilities of each event:
[tex]\[
P(A \text{ and } B) = P(A) \cdot P(B)
\][/tex]
Substituting the values of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]:
[tex]\[
P(A \text{ and } B) = \frac{5}{6} \cdot \frac{1}{6}
\][/tex]
Simplify the expression:
[tex]\[
P(A \text{ and } B) = \frac{5 \times 1}{6 \times 6} = \frac{5}{36}
\][/tex]
Therefore, the probability that both events A and B will occur is:
[tex]\[
P(A \text{ and } B) = \frac{5}{36}
\][/tex]
