To find the probability that both events A and B will occur, we need to follow these steps:
1. Define the Events:
- Event A: The first die does NOT land on 1.
- Event B: The second die does NOT land on 6.
2. Calculate the probabilities of each event:
- Probability of Event A:
A six-sided die has numbers 1 through 6. If the first die does NOT land on 1, it can land on 2, 3, 4, 5, or 6. That's 5 favorable outcomes out of the 6 possible outcomes.
Therefore, the probability of Event A is:
[tex]\[
P(A) = \frac{5}{6}
\][/tex]
- Probability of Event B:
A six-sided die also has numbers 1 through 6. If the second die does NOT land on 6, it can land on 1, 2, 3, 4, or 5. That's 5 favorable outcomes out of the 6 possible outcomes.
Therefore, the probability of Event B is:
[tex]\[
P(B) = \frac{5}{6}
\][/tex]
3. Calculate the probability that both events A and B will occur:
Since events A and B are independent (the outcome of one die does not affect the outcome of the other), the probability of both events occurring (P(A and B)) is given by the product of their individual probabilities:
[tex]\[
P(A \text{ and } B) = P(A) \cdot P(B)
\][/tex]
Plugging in the values, we get:
[tex]\[
P(A \text{ and } B) = \frac{5}{6} \cdot \frac{5}{6} = \frac{25}{36}
\][/tex]
Therefore, in simplest form, the probability that both events A and B will occur is:
[tex]\[
P(A \text{ and } B) = \frac{25}{36}
\][/tex]
So, the missing number in the fraction is 36.
