Problem 6 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The probability that at least one of the events \(A\) and \(B\) occurs is \(3 / 5\). If \(A\) and \(B\) occur simultaneously with probability \(1 / 5\), then \(P\left(A^{\prime}\right)+\) \(P\left(B^{\prime}\right)\) is (a) \(2 / 5\) (b) \(4 / 5\) (c) \(6 / 5\) (d) \(7 / 5\)

Step-by-Step Solution

Answer

The correct answer is (c) \(\frac{6}{5}\).

Step 1: Understanding the Problem

We are given the probability of at least one of the events \(A\) and \(B\) occurring, denoted as \(P(A \cup B) = \frac{3}{5}\). We are also given the probability that both \(A\) and \(B\) occur simultaneously, \(P(A \cap B) = \frac{1}{5}\). We need to find \(P(A') + P(B')\), where \(A'\) and \(B'\) are the complements of \(A\) and \(B\) respectively.

Step 2: Applying the Inclusion-Exclusion Principle

Apply the union formula for probability: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Given \(P(A \cup B) = \frac{3}{5}\) and \(P(A \cap B) = \frac{1}{5}\), substitute these into the formula. This gives us \(\frac{3}{5} = P(A) + P(B) - \frac{1}{5}\).

Step 3: Solving for \(P(A) + P(B)\)

Rearrange the equation \(\frac{3}{5} = P(A) + P(B) - \frac{1}{5}\) to find \(P(A) + P(B)\):\[P(A) + P(B) = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}.\] This is the combined probability of \(A\) and \(B\) occurring.

Step 4: Finding Complement Probabilities

We know that \(P(A') = 1 - P(A)\) and \(P(B') = 1 - P(B)\). We need \(P(A') + P(B')\), which is \((1 - P(A)) + (1 - P(B))\). Since \(P(A) + P(B) = \frac{4}{5}\), we have:\[(1 - P(A)) + (1 - P(B)) = 2 - (P(A) + P(B)) = 2 - \frac{4}{5}.\]

Step 5: Calculating the Final Expression

Now calculate \(2 - \frac{4}{5}\):\[2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5}.\]This is the value of \(P(A') + P(B')\).

Step 6: Choose the Correct Answer

Compare the solution with the given options. From the options provided, \(\frac{6}{5}\) corresponds to option (c).

Key Concepts

Complementary Events

Complementary events are an essential concept in probability theory. They express the idea of an event not happening. If you have an event \( A \), the complement event \( A' \) represents all outcomes where \( A \) does not occur. This is important in situations where you not only need to consider when something happens, but also when it doesn't. The sum of the probabilities of an event and its complement is always equal to 1.

Here's the key formula: - \( P(A') = 1 - P(A) \)
For example, if the probability of it raining today (event \( A \)) is 0.7, then the probability of it not raining (\( A' \)) is \( 1 - 0.7 = 0.3 \).

In the original exercise, once you find \( P(A) + P(B) \), you can easily calculate \( P(A') + P(B') \) using complementary probabilities.

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a crucial tool when calculating the probability of at least one event occurring. This principle helps us correct for overlap, ensuring we don't double-count occurrences.

The formula looks like this: - \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
When you want to find the probability of either event A or B happening, this formula gives a clear path, correcting any overlap by subtracting the probability of both events occurring simultaneously.

For example, in calculating \( P(A \cup B) \), if \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.2 \), then you calculate: - \( P(A \cup B) = 0.5 + 0.4 - 0.2 = 0.7 \)
In our original exercise, this principle helps derive the total probability of events \( A \) or \( B \) happening by considering the overlap \( P(A \cap B) \).

Simultaneous Events Probability

Understanding simultaneous events probability is vital when events can occur at the same time. Simultaneous events refer to the probability that both events \( A \) and \( B \) occur together, denoted by \( P(A \cap B) \).

This probability is part of solving problems with overlapping events. By calculating \( P(A \cap B) \), we understand how often both events intersect. This value becomes critical when applying the Inclusion-Exclusion Principle to avoid counting overlap twice.

In the example problem, \( P(A \cap B) = \frac{1}{5} \). This means there's a 20% chance both events occur at once. Knowing this probability is integral to finding the correct solution for \( P(A') + P(B') \) since it's needed to determine \( P(A \cup B) \).
To summarize, correctly identifying simultaneous events helps achieve precise probability calculations by accounting for overlaps in event occurrence.