Sure, let's solve this step-by-step:
1. Determine the Probability of Event A:
- Event A occurs if the first die lands on any of the numbers [tex]$1, 2, 3$[/tex], or [tex]$4$[/tex].
- There are [tex]$6$[/tex] possible outcomes when the first die is rolled: [tex]$1, 2, 3, 4, 5$[/tex], and [tex]$6$[/tex].
- The favorable outcomes for Event A are [tex]$1, 2, 3$[/tex], and [tex]$4$[/tex], which are [tex]$4$[/tex] outcomes.
- So, the probability of Event A, denoted as [tex]\( P(A) \)[/tex], is given by:
[tex]$
P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total possible outcomes}} = \frac{4}{6} = \frac{2}{3}
$[/tex]
In decimal form, [tex]\(\frac{2}{3} \approx 0.6667\)[/tex].
2. Determine the Probability of Event B:
- Event B occurs if the second die lands on [tex]$6$[/tex].
- There are [tex]$6$[/tex] possible outcomes when the second die is rolled: [tex]$1, 2, 3, 4, 5$[/tex], and [tex]$6$[/tex].
- The favorable outcome for Event B is [tex]$6$[/tex], which is [tex]$1$[/tex] outcome.
- So, the probability of Event B, denoted as [tex]\( P(B) \)[/tex], is given by:
[tex]$
P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total possible outcomes}} = \frac{1}{6}
$[/tex]
In decimal form, [tex]\(\frac{1}{6} \approx 0.1667\)[/tex].
3. Determine the Probability of Both Events Occurring:
- Since Events A and B are independent, the probability of both events occurring, denoted as [tex]\( P(A \text{ and } B) \)[/tex], is the product of their individual probabilities:
[tex]$
P(A \text{ and } B) = P(A) \cdot P(B) = \left( \frac{2}{3} \right) \cdot \left( \frac{1}{6} \right) = \frac{2}{18} = \frac{1}{9}
$[/tex]
In decimal form, [tex]\(\frac{1}{9} \approx 0.1111\)[/tex].
So, the probability that both events A and B will occur is [tex]\( \frac{1}{9} \)[/tex].
