To determine if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent or independent, we need to examine the probabilities involved and compare [tex]\( P(B) \)[/tex] with [tex]\( P(B \mid A) \)[/tex]:
1. Identify the events:
- Event [tex]\( A \)[/tex]: Rolling an even number on a six-sided die. The possible outcomes are [tex]\( \{2, 4, 6\} \)[/tex].
- Event [tex]\( B \)[/tex]: Rolling a 2 on a six-sided die. The only outcome is [tex]\( \{2\} \)[/tex].
2. Calculate the probabilities:
- The probability of event [tex]\( A \)[/tex] (rolling an even number) is given by:
[tex]\[
P(A) = \frac{\text{Number of favourable outcomes for event } A}{\text{Total number of outcomes}} = \frac{3}{6} = 0.5
\][/tex]
- The probability of event [tex]\( B \)[/tex] (rolling a 2) is given by:
[tex]\[
P(B) = \frac{\text{Number of favourable outcomes for event } B}{\text{Total number of outcomes}} = \frac{1}{6} \approx 0.16667
\][/tex]
- The probability of event [tex]\( B \)[/tex] given that event [tex]\( A \)[/tex] has occurred (rolling an even number and then checking if it's a 2) is:
[tex]\[
P(B \mid A) = \frac{\text{Number of favourable outcomes for event } B \text{ within } A}{\text{Number of outcomes for event } A} = \frac{1}{3} \approx 0.33333
\][/tex]
3. Determine if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent or independent:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if [tex]\( P(B) = P(B \mid A) \)[/tex].
- From the probabilities calculated, we have:
[tex]\[
P(B) \approx 0.16667 \quad \text{and} \quad P(B \mid A) \approx 0.33333
\][/tex]
- Since [tex]\( P(B) \neq P(B \mid A) \)[/tex], events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent.
Thus, the correct answer is:
Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent, because [tex]\( P(B) \neq P(B \mid A) \)[/tex]
