Probability Calculation for Events A and B

The problem asks us to find the probability of event A, denoted as P(A), given information about the probabilities of other related events: the probability of 'not B', the probability of the union of A and B, and the conditional probability of A given B.

Understanding the Given Probabilities

• P(not B) = 0.8: This is the probability that event B does not occur.
• P(A ∪ B) = 0.5: This is the probability that event A or event B (or both) occur.
• P(A|B) = 0.4: This is the probability that event A occurs given that event B has already occurred.

Applying Probability Rules

We will use fundamental probability rules to solve this problem:

• Complement Rule: The probability of an event occurring is 1 minus the probability of the event not occurring. Mathematically, $P(B) = 1 - P(\text{not } B)$.
• Conditional Probability Rule: The probability of event A occurring given event B has occurred is the probability of both events occurring divided by the probability of B. Mathematically, $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(A \cap B)$ is the probability of both A and B occurring.
• Addition Rule for Probabilities: The probability of the union of two events A and B is the sum of their individual probabilities minus the probability of their intersection. Mathematically, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Step-by-Step Calculation of P(A)

Let's use the given information and the rules to find P(A).

Step 1: Find P(B)

We are given $P(\text{not } B) = 0.8$. Using the complement rule:

So, the probability of event B occurring is 0.2.

Step 2: Find P(A ∩ B)

We are given $P(A|B) = 0.4$ and we found $P(B) = 0.2$. Using the conditional probability rule:

We can rearrange this formula to solve for $P(A \cap B)$:

The probability of both A and B occurring is 0.08.

Step 3: Find P(A)

We are given $P(A \cup B) = 0.5$, we found $P(B) = 0.2$, and we found $P(A \cap B) = 0.08$. Using the addition rule for probabilities:

Substitute the known values into the equation:

Simplify the right side of the equation:

Now, isolate P(A) by subtracting 0.12 from both sides:

Conclusion

Based on the given probabilities and applying the fundamental rules of probability, the probability of event A occurring, P(A), is 0.38.

Revision Table: Key Probability Formulas

Additional Information: Types of Events

Understanding different types of events is crucial in probability:

• Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, $P(A|B) = P(A)$ or $P(B|A) = P(B)$, which implies $P(A \cap B) = P(A)P(B)$.
• Mutually Exclusive Events: Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time. Mathematically, $P(A \cap B) = 0$. For mutually exclusive events, the addition rule simplifies to $P(A \cup B) = P(A) + P(B)$. In this problem, since $P(A \cap B) = 0.08 \neq 0$, events A and B are not mutually exclusive.
• Dependent Events: Events that are not independent are dependent. The conditional probability $P(A|B)$ is generally not equal to $P(A)$ for dependent events. In this problem, $P(A|B) = 0.4$ and we found $P(A) = 0.38$. Since $0.4 \neq 0.38$, A and B are dependent events.