Problem 5 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(A\) and \(B\) are two events such that \(P(A)=\) \(0.42, P(B)=0.48\) and \(P(A B)=0.16\), find \(P(A+B)\) [MP-1998]

Step-by-Step Solution

Answer

The probability of event \(A\) or \(B\) occurring, \(P(A \cup B)\), is 0.74.

Step 1: Understand the Problem

We are given two events, \(A\) and \(B\), with their respective probabilities: \(P(A) = 0.42\), \(P(B) = 0.48\), and the probability of both events occurring together: \(P(A \cap B) = 0.16\). We are asked to find the probability of either event \(A\) or event \(B\) occurring, denoted as \(P(A \cup B)\).

Step 2: Use the Inclusion-Exclusion Principle

According to the formula for the probability of the union of two events in probability theory, we have:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]This formula helps adjust for the fact that \(P(A)\) and \(P(B)\) count the overlap twice.

Step 3: Substitute the Given Values

Substitute \(P(A) = 0.42\), \(P(B) = 0.48\), and \(P(A \cap B) = 0.16\) into the formula:\[P(A \cup B) = 0.42 + 0.48 - 0.16\]

Step 4: Calculate the Probability of the Union

Perform the calculations:\[P(A \cup B) = 0.42 + 0.48 - 0.16 = 0.74\]

Step 5: Interpret the Result

The probability of either event \(A\) or event \(B\) happening is \(0.74\). This means there is a 74% chance that at least one of the events occurs.

Key Concepts

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a fundamental concept in probability theory and combinatorics. It is used to calculate the probability or count of the union of multiple sets or events. When dealing with two or more events, we need to be careful not to double-count elements that are in more than one event. This principle adjusts for the overlap between events.

This principle helps by providing a systematic way to include the probabilities of individual events, while excluding the overlap counted twice. In mathematical terms, for two events \(A\) and \(B\), the probability of their union can be calculated as:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Here, \(P(A \cap B)\) is subtracted because the overlap between \(A\) and \(B\) gets counted twice if we just add \(P(A)\) and \(P(B)\). This simple intuition helps in dealing with multiple overlapping events, ensuring that each part of the combined scenario is counted just once.

Probability of Union

The probability of the union of two events, often noted as \(P(A \cup B)\), signifies the likelihood of either event \(A\), event \(B\), or both occurring. The key here is understanding that the union encompasses all outcomes that satisfy at least one of the involved events.

Using the inclusion-exclusion principle formula explained earlier, we see how the formula accounts for the intersection—where both events occur simultaneously—adjusting this in the sum.

For example, if the probability of rolling a dice and getting a 2 is 1/6, and the same goes for rolling a 4, the probability of rolling either a 2 or a 4 isn't just their sum (1/6 + 1/6 = 1/3), because there isn't any overlap when dealing with a die. However, in cases where overlap exists, it's crucial to subtract the intersecting probability; in our exercise example:

\[ P(A \cup B) = 0.42 + 0.48 - 0.16 = 0.74 \]

This means that there is a 74% chance of occurrence of either \(A\), \(B\), or both.

Events in Probability

Events in probability refer to the possible outcomes or occurrences that can happen in a random experiment. An event could be as simple as getting a heads when flipping a coin, or more complex like having the exact combination of cards drawn from a deck.

Events can be categorized in several ways, most notably as mutually exclusive or inclusive. Mutually exclusive events can't occur at the same time (like rolling a 2 or a 3 on a single dice roll). Non-mutually exclusive events, like picking a red card or a face card from a deck, can have overlap (a red face card).

In probability, each event is associated with a probability value between 0 and 1, indicating the likelihood of the event occurring. When calculating probabilities, it's crucial to understand how events interact—whether they can occur simultaneously or affect each other.

The understanding of how to handle these interactions underpins more sophisticated calculations, such as using the inclusion-exclusion principle for computing the probability of the union of overlapping events.