Two events A and B are such that P(not B) = 0.8, P(A ∪ B) = 0.5 and P(A|B) = 0.4. Then P(A) is equal to
0.38
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.
| Probability | Value |
|---|---|
| P(not B) | 0.8 (Given) |
| P(A ∪ B) | 0.5 (Given) |
| P(A|B) | 0.4 (Given) |
| P(B) | 0.2 (Calculated from P(not B)) |
| P(A ∩ B) | 0.08 (Calculated from P(A|B) and P(B)) |
| P(A) | 0.38 (Calculated from P(A ∪ B), P(B), and P(A ∩ B)) |
Revision Table: Key Probability Formulas
| Formula Name | Formula | Explanation |
|---|---|---|
| Complement Rule | $P(E^c) = 1 - P(E)$ | Probability of event E not happening ($E^c$) is 1 minus probability of E happening. |
| Conditional Probability | $P(E|F) = \frac{P(E \cap F)}{P(F)}$ | Probability of E given F has occurred is probability of E and F both occurring, divided by probability of F. Valid when $P(F) > 0$. |
| Addition Rule (Union) | $P(E \cup F) = P(E) + P(F) - P(E \cap F)$ | Probability of E or F (or both) occurring is sum of individual probabilities minus probability of both occurring. |
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.
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