Problem 12 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Let \(A\) and \(B\) be two events such that \(P(A)=0.3\) and \(P(A \cup B)=0.8\). If \(A\) and \(B\) are independent events, then \(P(B)\) is (a) \(5 / 6\) (b) \(5 / 7\) (c) \(3 / 5\) (d) \(2 / 5\)

Step-by-Step Solution

Answer

The probability \( P(B) \) is option (b) \( \frac{5}{7} \).

Step 1: Identifying Given Data

Let's start by noting the given information: \( P(A) = 0.3 \) and \( P(A \cup B) = 0.8 \). We also know that events \( A \) and \( B \) are independent.

Step 2: Understanding Independence

For independent events, the probability of \( A \cap B \) is \( P(A \cap B) = P(A) \times P(B) \). This property will be useful for calculating probabilities.

Step 3: Using the Formula for Union of Events

The formula for the probability of the union of two events is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). We will use this formula and the independence information to find \( P(B) \).

Step 4: Substituting Known Values

Substitute the known values into the union formula: \( 0.8 = 0.3 + P(B) - (0.3 \times P(B)) \).

Step 5: Simplifying the Equation

Rearrange and simplify the equation: \[ 0.8 = 0.3 + P(B) - 0.3P(B) \] \[ 0.8 = 0.3 + P(B)(1 - 0.3) \] \[ 0.8 = 0.3 + 0.7P(B) \].

Step 6: Solving for \( P(B) \)

Rearrange the simplified equation to solve for \( P(B) \): \[ 0.8 - 0.3 = 0.7P(B) \] \[ 0.5 = 0.7P(B) \] \[ P(B) = \frac{0.5}{0.7} = \frac{5}{7} \].

Key Concepts

Independent Events

In probability theory, events are said to be independent if the occurrence or non-occurrence of one event does not affect the probability of occurrence of another event. This characteristic is fundamental as it allows us to determine complicated probabilities simplistically.
For example, if you flip a fair coin twice, the outcome of the first flip does not influence the outcome of the second flip. Each flip is independent.
When dealing with independent events, there is a special rule we can use: the probability of both events occurring (happening together) is simply the product of their individual probabilities.

• If events \( A \) and \( B \) are independent, then \( P(A \cap B) = P(A) \times P(B) \).
• This formula simplifies the calculations significantly when dealing with these types of events.

Understanding when events are independent can make problem-solving easier and quicker, by breaking down complex problems into simpler calculations.

Probability of Union

The probability of the union of two events, often represented as \( P(A \cup B) \), is a measure of the likelihood that at least one of the events will occur. For example, if you're drawing a card from a deck, the probability of drawing a heart or a face card can be computed using the formula for union.
For any two events \( A \) and \( B \), the formula is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]This equation accounts for the fact that if the events can occur simultaneously, we must subtract that overlap once because it's been counted twice.

• The overlap, \( P(A \cap B) \), is subtracted because it's included in both \( P(A) \) and \( P(B) \).
• This formula is essential in finding probabilities of either event occurring when they are not mutually exclusive.

In our exercise scenario, by using this formula, we can solve for one of the unknown probabilities if others are known.

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It provides the tools we need to quantify and analyze events' likelihood in various contexts, from simple games of chance to complex scientific experiments.
This field is built on some fundamental principles, including: **events** (outcomes or sets of outcomes in a probability space) and **probabilities** (numbers between 0 and 1 associated with events representing the event's likelihood).

• A probability of 0 means the event is impossible, while a probability of 1 indicates certainty.
• Common operations in probability include finding the probability of unions, intersections, and complements of events.

Concepts like independent events and formulas like those for the union of events arise from these basic tenets.
Understanding probability theory helps us develop a deeper intuition about how likely events are and enables us to make predictions and informed decisions based on data and reasoning.