To solve the problem of finding the probability that event $A$ or $B$ will occur, denoted as $P(A \cup B)$, we use the formula for the union of two events:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

From the Venn diagram:

• $P(A)$, the probability of event $A$, is given by adding the probabilities inside circle $A$, which is $\frac{4}{15} + \frac{1}{15} = \frac{5}{15}$.
• $P(B)$, the probability of event $B$, is given by adding the probabilities inside circle $B$, which is $\frac{2}{15} + \frac{1}{15} = \frac{3}{15}$.
• $P(A \cap B)$, the probability that both $A$ and $B$ occur (the overlap), is $\frac{1}{15}$.

Now apply the formula:

$P(A \cup B) = \frac{5}{15} + \frac{3}{15} - \frac{1}{15}$ $P(A \cup B) = \frac{7}{15}$

Thus, the probability that event $A$ or $B$ will occur is:

$P(A \cup B) = \frac{7}{15}$

Would you like more details or have any questions?

Related Questions:

1. What is the probability of event $A$ happening but not $B$?
2. How does the formula for $P(A \cup B)$ change if the events are mutually exclusive?
3. What is the probability that neither event $A$ nor event $B$ will occur?
4. Can you extend this probability concept to three events?
5. How is the complement rule applied in this case?

Tip:

In probability, always remember to subtract the overlap $P(A \cap B)$ when calculating $P(A \cup B)$, to avoid double-counting.