Problem 15 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(A\) and \(B\) are two events such that \(P(A \cup B)+\) \(P(A \cap B)=7 / 8\) and \(P(A)=2 P(B)\), then \(P(A)\) is (a) \(7 / 12\) (b) \(7 / 24\) (c) \(5 / 12\) (d) \(17 / 24\)

Step-by-Step Solution

Answer

The probability \(P(A)\) is \(\frac{7}{12}\).

Step 1: Identify Given Information

We are given the probability equation: \(P(A \cup B) + P(A \cap B) = \frac{7}{8}\). Also, \(P(A) = 2P(B)\).

Step 2: Apply the Probability Formula for Union

Recall the formula for the probability of the union of two events: \[P(A \cup B) = P(A) + P(B) - P(A \cap B).\]We can substitute and rewrite it in terms of known quantities.

Step 3: Express in Terms of Known Probabilities

Replace \(P(A \cup B)\) in the given equation to get: \[P(A) + P(B) - P(A \cap B) + P(A \cap B) = \frac{7}{8}.\] This simplifies to: \[P(A) + P(B) = \frac{7}{8}.\]

Step 4: Substitute \(P(A) = 2P(B)\)

Substitute \(P(A) = 2P(B)\) in the simplified equation from Step 3:\[2P(B) + P(B) = \frac{7}{8}.\]This simplifies to \[3P(B) = \frac{7}{8}.\]

Step 5: Solve for \(P(B)\)

To solve for \(P(B)\), divide both sides of the equation by 3:\[P(B) = \frac{7}{24}.\]

Step 6: Find \(P(A)\)

Using \(P(A) = 2P(B)\), substitute the value of \(P(B)\):\[P(A) = 2 \times \frac{7}{24} = \frac{14}{24} = \frac{7}{12}.\]

Key Concepts

Probability of Union

In probability theory, understanding the concept of the union of two events is crucial. The union of two events, say events \( A \) and \( B \), is represented as \( A \cup B \). It encompasses all possible outcomes that are in either \( A \), \( B \), or both. Essentially, if you think of events as circles in a Venn diagram, the union is all the areas covered by both circles.
When calculating the probability of a union, we want to know the probability that at least one of the events will occur. To find this, we use the formula:

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

This formula helps to ensure we don’t count the overlap (i.e., the intersection where both \( A \) and \( B \) happen) twice. It’s a versatile tool whenever discussing probabilities of multiple events simultaneously.

Intersection of Events

The intersection of two events represents scenarios where both events occur simultaneously. In notational terms, this is expressed as \( A \cap B \). If you visualize it with a Venn diagram, the intersection is the area common to both circles of \( A \) and \( B \).
To find the probability of this intersection, it's often given or needs to be derived based on other known probabilities. For example, in our step-by-step solution, the intersection probability was part of the given combined probability equation. Recognizing the intersection is key because it ensures that overlapping probabilities are handled correctly, which is vital in precise probability calculations.
It's especially important in cases where events \( A \) and \( B \) are not mutually exclusive, and need to be carefully accounted for using the proper probability formulas.

Probability Formula

The foundation of tackling probability problems is a good grasp of the fundamental probability formulas. In our example, we deal with the formula for the union of two events, but also initially leverage conditional knowledge like \( P(A) = 2P(B) \). Formulas in probability provide the framework to correctly allocate likelihoods to various outcomes.
The union formula as revisited is:

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

This formula ensures that overlaps are not inflated by considering the intersection term. Moreover, equations such as \( P(A) = 2P(B) \) imposed complementariness between the events allowing more straightforward solutions through substitutions. Knowing how and when to apply these formulas is pivotal for determining the precise probabilities of composite events and obtaining the solution to problems like the one initially presented.