For any two events, A and B, it is given that P(A and B) = 0,12, P(A or B) = 0,83 and P(B) = 4 P(A). Are events A and B mutually exclusive? Justify your answer. Calculate P(B). Calculate P(not A).

Events A and B are mutually exclusive if the probability of both events occurring, $P(A \text{ and } B)$, is 0.

We are given $P(A \text{ and } B) = 0.12$.
Since $P(A \text{ and } B) = 0.12 \neq 0$, the events are not mutually exclusive.

No, events A and B are not mutually exclusive.
This is because $P(A \text{ and } B) = 0.12$, which is not equal to 0.

9.2.2 Calculate P(B).

The Addition Rule states:

We are given $P(A \text{ or } B) = 0.83$ and $P(A \text{ and } B) = 0.12$. We also know $P(B) = 4P(A)$, which implies $P(A) = \frac{P(B)}{4}$.

First, add 0.12 to both sides:

Combine the terms with $P(B)$:

Multiply both sides by $\frac{4}{5}$ to find $P(B)$:

9.2.3 Calculate P(not A).

From the previous calculation, we found $P(B) = 0.76$.

The probability of an event not occurring is 1 minus the probability of the event occurring: