Understanding the Problem: Calculating Conditional Probability

The question asks us to find the conditional probability P(A|B) given information about the probabilities of events A and B, and the relationship between their complements A̅ and B̅. We are given:

• A̅ and B̅ are mutually exclusive events.
• P(A) = 0.5
• P(B) = 0.6

Our goal is to calculate P(A|B), which is the probability of event A occurring given that event B has already occurred.

Mutually Exclusive Complements and Their Implication

Two events are mutually exclusive if they cannot happen at the same time. This means the probability of their intersection is zero.

• A̅ and B̅ are mutually exclusive, so P(A̅ ∩ B̅) = 0.

Now, let's consider what the intersection of the complements, A̅ ∩ B̅, represents. According to De Morgan's Laws, the intersection of the complements of two events is the complement of their union:

\(\text{A̅} \cap \text{B̅} = \overline{(\text{A} \cup \text{B})}\)

Therefore, if P(A̅ ∩ B̅) = 0, it means:

\(P(\overline{\text{A} \cup \text{B}}) = 0\)

The probability of the complement of an event is 1 minus the probability of the event itself. So, \(P(\overline{\text{A} \cup \text{B}}) = 1 - P(\text{A} \cup \text{B})\).

Since \(P(\overline{\text{A} \cup \text{B}}) = 0\), we have:

\(1 - P(\text{A} \cup \text{B}) = 0\)

This implies:

\(P(\text{A} \cup \text{B}) = 1\)

This is a crucial piece of information derived from the fact that A̅ and B̅ are mutually exclusive.

Using the Union Formula to Find Intersection

The formula for the probability of the union of two events A and B is:

\(P(\text{A} \cup \text{B}) = P(\text{A}) + P(\text{B}) - P(\text{A} \cap \text{B})\)

We know P(A ∪ B) = 1, P(A) = 0.5, and P(B) = 0.6. We can use this formula to find P(A ∩ B), the probability of the intersection of A and B.

Substitute the known values into the formula:

\(1 = 0.5 + 0.6 - P(\text{A} \cap \text{B})\)

Calculate the sum of P(A) and P(B):

\(1 = 1.1 - P(\text{A} \cap \text{B})\)

Now, solve for P(A ∩ B):

\(P(\text{A} \cap \text{B}) = 1.1 - 1\)

\(P(\text{A} \cap \text{B}) = 0.1\)

So, the probability that both event A and event B occur is 0.1.

Calculating Conditional Probability P(A|B)

The formula for conditional probability P(A|B) is:

\(P(\text{A}|\text{B}) = \frac{P(\text{A} \cap \text{B})}{P(\text{B})}\)

We have found P(A ∩ B) = 0.1 and we are given P(B) = 0.6.

Substitute these values into the conditional probability formula:

\(P(\text{A}|\text{B}) = \frac{0.1}{0.6}\)

Simplify the fraction:

\(P(\text{A}|\text{B}) = \frac{\frac{1}{10}}{\frac{6}{10}}\)

\(P(\text{A}|\text{B}) = \frac{1}{10} \times \frac{10}{6}\)

\(P(\text{A}|\text{B}) = \frac{1}{6}\)

Step-by-Step Solution Summary

1. Given A̅ and B̅ are mutually exclusive, P(A̅ ∩ B̅) = 0.
2. Using De Morgan's Law, P((A ∪ B)̅) = P(A̅ ∩ B̅) = 0.
3. Since P((A ∪ B)̅) = 0, it implies P(A ∪ B) = 1 - 0 = 1.
4. Using the union formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
5. Substitute values: 1 = 0.5 + 0.6 - P(A ∩ B).
6. Solve for P(A ∩ B): P(A ∩ B) = 1.1 - 1 = 0.1.
7. Using the conditional probability formula P(A|B) = P(A ∩ B) / P(B).
8. Substitute values: P(A|B) = 0.1 / 0.6.
9. Simplify: P(A|B) = 1/6.

The value of P(A|B) is \(\frac{1}{6}\).

Revision Table: Key Probabilities

Additional Information: Probability Concepts

Understanding key probability concepts is essential for solving such problems. Here are some related terms:

• Mutually Exclusive Events: Two events E1 and E2 are mutually exclusive if they cannot occur at the same time, i.e., E1 ∩ E2 = ∅ (the empty set). This means P(E1 ∩ E2) = 0.
• Conditional Probability: The probability of an event A occurring given that another event B has already occurred is denoted by P(A|B) and is calculated as \(P(\text{A}|\text{B}) = \frac{P(\text{A} \cap \text{B})}{P(\text{B})}\), provided P(B) > 0.
• De Morgan's Laws: These are rules relating the intersection and union of sets and their complements. For events A and B:
  • \(\overline{\text{A} \cup \text{B}} = \text{A̅} \cap \text{B̅}\)
  • \(\overline{\text{A} \cap \text{B}} = \text{A̅} \cup \text{B̅}\)
• Probability of Union: The probability that either event A or event B (or both) occurs is given by \(P(\text{A} \cup \text{B}) = P(\text{A}) + P(\text{B}) - P(\text{A} \cap \text{B})\). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

In this problem, the condition that the complements A̅ and B̅ are mutually exclusive provides a strong constraint on the relationship between events A and B, specifically forcing their union A ∪ B to be the entire sample space (probability 1).