Problem 1 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(4 P(A)=6 P(B)=10(A \cap B)=1\), then \(P(B / \mathrm{A})=\) (a) \(2 / 5\) (b) \(3 / 5\) (c) \(7 / 10\) (d) \(19 / 60\)

Step-by-Step Solution

Answer

The probability \(P(B|A)\) is \(\frac{2}{5}\), which corresponds to option (a).

Step 1: Understand Given Equations

The problem states that \(4 P(A) = 6 P(B) = 10 P(A \cap B) = 1\). This indicates each of these expressions is equal to 1, allowing us to set up the following equations: \(P(A) = \frac{1}{4}\), \(P(B) = \frac{1}{6}\), and \(P(A \cap B) = \frac{1}{10}\).

Step 2: Apply Conditional Probability Formula

We need to find \(P(B|A)\), which is the probability of \(B\) given \(A\). The formula for conditional probability is \(P(B|A) = \frac{P(A \cap B)}{P(A)}\).

Step 3: Substitute Values into Formula

Substitute the known values into the formula: \(P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{1}{10}}{\frac{1}{4}}\).

Step 4: Simplify the Expression

Simplify the fraction \(\frac{\frac{1}{10}}{\frac{1}{4}}\) by multiplying by the reciprocal: \(\frac{1}{10} \times \frac{4}{1} = \frac{4}{10}\).

Step 5: Final Simplification

Simplify \(\frac{4}{10}\) to \(\frac{2}{5}\). Therefore, \(P(B|A) = \frac{2}{5}\).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood or chance of different outcomes happening. It provides the mathematical foundation for assessing risks in various fields, including science, engineering, and finance. In probability theory, events are typically assigned a probability value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

To understand probability theory, consider a simple example, such as flipping a coin. The probability of getting heads or tails is equal, giving each outcome a probability of 0.5. In the context of the exercise, we look at different probabilities for events and combinations of events. For instance:

• Event A could represent rolling a 4 on a six-sided die.
• Event B could represent rolling an even number.

The principle of probability theory allows us to calculate how likely it is for event A or event B, or both, to occur.

Set Operations in Probability

In probability, set operations approach is vital for understanding the relationships between different events. You might have heard of concepts like union, intersection, and complement.

- **Union of Sets:** This operation, denoted as \(A \cup B\), represents all outcomes that are in event A, event B, or in both. It's like combining the results of two events.- **Intersection of Sets:** Represented by \(A \cap B\), this is the set of outcomes that are common to both events A and B. In the given exercise, \(P(A \cap B)\) is the probability of both events A and B occurring simultaneously.- **Complementary Sets:** The complement of event A, denoted \(A^c\), includes all possible outcomes that are not in A.These set operations enable more complex probability calculations by linking them with familiar set manipulation approaches, similar to those used in Venn diagrams.

Algebraic Equations in Probability

When studying probability, you often need to use algebraic equations to solve problems. These equations help us decipher relationships between different probabilities. In this exercise, you're given equations like

• \(4P(A) = 6P(B) = 10P(A \cap B) = 1\)

This shows that all these probabilities are interconnected and equal to one another through their manipulated algebraic representations.

To find conditional probabilities, like \(P(B|A)\), you use the relationship \(P(B|A) = \frac{P(A \cap B)}{P(A)}\). This equation describes the probability of event B occurring given that event A has occurred.

Substitute the values from the given equations into this conditional formula. It's similar to substituting values into algebraic formulas; just substitute \(P(A \cap B)\) and \(P(A)\) with their respective values to find \(P(B|A)\). Solving these equations requires skills in manual manipulation and simplification of fractions, which is part of basic algebra.