Answer :
Let's break down and solve the problem with the given events and probabilities step by step.
### Step 1: Calculate the Probability of Event A
Event A: The first die lands on 1 or 2.
A standard six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- Number of favorable outcomes for Event A (landing on 1 or 2) = 2
- Total number of possible outcomes for the first die = 6
Therefore, the probability of Event A occurring is:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes for Event A}}{\text{Total number of possible outcomes for the first die}} = \frac{2}{6} = \frac{1}{3} \][/tex]
### Step 2: Calculate the Probability of Event B
Event B: The second die lands on 5.
Again, a standard six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- Number of favorable outcomes for Event B (landing on 5) = 1
- Total number of possible outcomes for the second die = 6
Therefore, the probability of Event B occurring is:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes for Event B}}{\text{Total number of possible outcomes for the second die}} = \frac{1}{6} \][/tex]
### Step 3: Calculate the Joint Probability of Both Events A and B Occurring
Since the events are independent, the probability of both events occurring (Event A and Event B) is:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Using the probabilities calculated:
[tex]\[ P(A \text{ and } B) = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{6}\right) \][/tex]
[tex]\[ P(A \text{ and } B) = \frac{1}{3} \cdot \frac{1}{6} = \frac{1}{18} \][/tex]
### Conclusion
Thus, the probability that both events will occur (the first die lands on 1 or 2 and the second die lands on 5) is:
[tex]\[ P(A \text{ and } B) = \frac{1}{18} \][/tex]
### Final Answer in Simplest Form
The final answer is:
[tex]\[ \boxed{\frac{1}{18}} \][/tex]
### Step 1: Calculate the Probability of Event A
Event A: The first die lands on 1 or 2.
A standard six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- Number of favorable outcomes for Event A (landing on 1 or 2) = 2
- Total number of possible outcomes for the first die = 6
Therefore, the probability of Event A occurring is:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes for Event A}}{\text{Total number of possible outcomes for the first die}} = \frac{2}{6} = \frac{1}{3} \][/tex]
### Step 2: Calculate the Probability of Event B
Event B: The second die lands on 5.
Again, a standard six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- Number of favorable outcomes for Event B (landing on 5) = 1
- Total number of possible outcomes for the second die = 6
Therefore, the probability of Event B occurring is:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes for Event B}}{\text{Total number of possible outcomes for the second die}} = \frac{1}{6} \][/tex]
### Step 3: Calculate the Joint Probability of Both Events A and B Occurring
Since the events are independent, the probability of both events occurring (Event A and Event B) is:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Using the probabilities calculated:
[tex]\[ P(A \text{ and } B) = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{6}\right) \][/tex]
[tex]\[ P(A \text{ and } B) = \frac{1}{3} \cdot \frac{1}{6} = \frac{1}{18} \][/tex]
### Conclusion
Thus, the probability that both events will occur (the first die lands on 1 or 2 and the second die lands on 5) is:
[tex]\[ P(A \text{ and } B) = \frac{1}{18} \][/tex]
### Final Answer in Simplest Form
The final answer is:
[tex]\[ \boxed{\frac{1}{18}} \][/tex]
