For two mutually exclusive events A and B, P(A) = 0.2 and P (A̅ ∩ B) = 0.3. What is P (A|(A ∪ B)) equal to?
Understanding Probability with Mutually Exclusive Events
The question asks us to find the conditional probability \(P(A | (A \cup B))\) for two mutually exclusive events A and B, given \(P(A) = 0.2\) and \(P(A̅ \cap B) = 0.3\).
Let's break down the information given and the terms used:
- Mutually Exclusive Events: Two events A and B are mutually exclusive if they cannot occur at the same time. This means their intersection is empty, i.e., \(A \cap B = \emptyset\). Consequently, the probability of their intersection is zero: \(P(A \cap B) = 0\).
- Given Probabilities:
- \(P(A) = 0.2\)
- \(P(A̅ \cap B) = 0.3\). This represents the probability that event B occurs and event A does not occur.
Analyzing \(P(A̅ \cap B)\) for Mutually Exclusive Events
Since A and B are mutually exclusive, if event B occurs, event A cannot occur. This means that the set of outcomes where B happens and A does not happen (\(A̅ \cap B\)) is simply the set of outcomes where B happens, because any outcome in B is automatically not in A. Therefore, \(A̅ \cap B = B\).
So, the given probability \(P(A̅ \cap B) = 0.3\) implies \(P(B) = 0.3\).
Let's summarize the probabilities we have:
| Event | Probability |
|---|---|
| A | \(P(A) = 0.2\) |
| B | \(P(B) = 0.3\) |
| A and B (Intersection) | \(P(A \cap B) = 0\) (since mutually exclusive) |
Calculating \(P(A \cup B)\)
For mutually exclusive events A and B, the probability of their union (A or B occurring) is the sum of their individual probabilities:
\(P(A \cup B) = P(A) + P(B)\)
Substituting the values we have:
\(P(A \cup B) = 0.2 + 0.3 = 0.5\)
Calculating the Conditional Probability \(P(A | (A \cup B))\)
The formula for conditional probability \(P(X | Y)\) is \(\frac{P(X \cap Y)}{P(Y)}\). In this question, \(X = A\) and \(Y = A \cup B\). So, we need to calculate:
\(P(A | (A \cup B)) = \frac{P(A \cap (A \cup B))}{P(A \cup B)}\)
Let's simplify the term \(A \cap (A \cup B)\). The intersection of event A with the union of A and B contains all outcomes that are in A AND are also in either A or B. The outcomes that satisfy this condition are simply the outcomes that are in A. Therefore, \(A \cap (A \cup B) = A\).
Now, substitute this back into the conditional probability formula:
\(P(A | (A \cup B)) = \frac{P(A)}{P(A \cup B)}\)
We have \(P(A) = 0.2\) and we calculated \(P(A \cup B) = 0.5\).
Substituting these values:
\(P(A | (A \cup B)) = \frac{0.2}{0.5} = \frac{\frac{2}{10}}{\frac{5}{10}} = \frac{2}{5}\)
Thus, the conditional probability \(P(A | (A \cup B))\) is \(\frac{2}{5}\).
Step-by-Step Solution Summary
- Identify that A and B are mutually exclusive, so \(P(A \cap B) = 0\).
- Use \(P(A̅ \cap B) = 0.3\) to find \(P(B)\). Since A and B are mutually exclusive, \(A̅ \cap B = B\), so \(P(B) = 0.3\).
- Calculate \(P(A \cup B)\) using the formula for mutually exclusive events: \(P(A \cup B) = P(A) + P(B) = 0.2 + 0.3 = 0.5\).
- Simplify the intersection term in the conditional probability formula: \(A \cap (A \cup B) = A\).
- Apply the conditional probability formula: \(P(A | (A \cup B)) = \frac{P(A \cap (A \cup B))}{P(A \cup B)} = \frac{P(A)}{P(A \cup B)}\).
- Substitute the known values: \(P(A | (A \cup B)) = \frac{0.2}{0.5} = \frac{2}{5}\).
Revision Table: Probability Concepts
| Concept | Definition/Formula | Notes |
|---|---|---|
| Mutually Exclusive Events A, B | \(A \cap B = \emptyset\) | Cannot happen simultaneously |
| Probability of Intersection (Mutually Exclusive) | \(P(A \cap B) = 0\) | Zero probability of both occurring |
| Probability of Union (Mutually Exclusive) | \(P(A \cup B) = P(A) + P(B)\) | Sum of individual probabilities |
| Conditional Probability \(P(A | B)\) | \(\frac{P(A \cap B)}{P(B)}\) | Probability of A given B has occurred |
| Complement of Event A (\(A̅\)) | Outcomes not in A | \(P(A̅) = 1 - P(A)\) |
| Intersection of Complement and Event (\(A̅ \cap B\)) | Outcomes in B but not in A | If A and B are mutually exclusive, \(A̅ \cap B = B\) |
Additional Information: Set Operations in Probability
In probability, events are treated as sets of outcomes. Understanding basic set operations helps in solving problems.
- Union (\(\cup\)): \(A \cup B\) is the event where A occurs, or B occurs, or both occur.
- Intersection (\(\cap\)): \(A \cap B\) is the event where both A and B occur.
- Complement (\(A̅\)): \(A̅\) is the event where A does not occur.
For any two events A and B, the general formula for the probability of their union is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). When events are mutually exclusive, \(P(A \cap B) = 0\), simplifying the formula to \(P(A \cup B) = P(A) + P(B)\).
The term \(A̅ \cap B\) represents the part of event B that does not overlap with event A. This is often written as \(B \setminus A\) (B minus A). For mutually exclusive events, there is no overlap (\(A \cap B = \emptyset\)), so the part of B that doesn't overlap with A is simply all of B (\(B \setminus A = B\)). This confirms why \(A̅ \cap B = B\) for mutually exclusive A and B.
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