Problem 19 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Three groups \(A, B, C\) are competing for positions on the Board of Directors of a company. The probabilities of their winning are \(0.5\), \(0.3,0.2\), respectively. If the group \(A\) wins, the probability of introducing a new product is \(0.7\) and the corresponding probabilities for group \(B\) and \(C\) are \(0.6\) and \(0.5\), respectively. The probability that the new product will be introduced is (a) \(0.18\) (b) \(0.35\) (c) \(0.10\) (d) \(0.63\)

Step-by-Step Solution

Answer

The probability that the new product will be introduced is 0.63, option (d).

Step 1: Identify Given Probabilities

We are given the probabilities of each group winning the Board elections: - Probability of group \( A \) winning, \( P(A) = 0.5 \) - Probability of group \( B \) winning, \( P(B) = 0.3 \) - Probability of group \( C \) winning, \( P(C) = 0.2 \).Additionally, we have the following conditional probabilities for introducing a new product:- If \( A \) wins, \( P(\text{new product} | A) = 0.7 \)- If \( B \) wins, \( P(\text{new product} | B) = 0.6 \)- If \( C \) wins, \( P(\text{new product} | C) = 0.5 \).

Step 2: Use Total Probability Theorem

We will use the law of total probability to compute the overall probability of introducing a new product:\[ P(\text{new product}) = P(\text{new product} | A) \cdot P(A) + P(\text{new product} | B) \cdot P(B) + P(\text{new product} | C) \cdot P(C) \]

Step 3: Calculate Individual Terms

Calculate the contribution of each group to the probability of the new product being introduced:- For group \( A \): \( 0.7 \times 0.5 = 0.35 \)- For group \( B \): \( 0.6 \times 0.3 = 0.18 \)- For group \( C \): \( 0.5 \times 0.2 = 0.10 \)

Step 4: Sum the Contributions

Add the contributions from all groups to find the total probability:\[ P(\text{new product}) = 0.35 + 0.18 + 0.10 = 0.63 \]

Step 5: Find the Correct Option

Compare the calculated probability to the provided options: - (a) 0.18 - (b) 0.35 - (c) 0.10 - (d) 0.63 The correct option is (d) 0.63.

Key Concepts

Conditional Probability

Conditional probability is a cornerstone of understanding how likelihoods can change based on specific conditions or events occurring. In simple terms, it answers the question, "Given that this happens, what is the chance that happens?"

For this exercise, we identified the conditional probabilities for introducing a new product based on the election outcomes:

• If group A wins, the conditional probability of introducing the product is 0.7.
• If group B wins, the probability drops to 0.6.
• For group C, the probability is 0.5.

These values illustrate how the likelihood of the new product launches is influenced directly by which group wins the election. Understanding these dependencies allows us to apply more advanced probability concepts to solve related problems.

Law of Total Probability

The Law of Total Probability is a fundamental rule used to calculate the probability of an entire event by integrating its conditional probabilities over a partition of sample spaces.

In this scenario, the partitions are the possible outcomes of the election (which group wins). Let's break it down:

• The overall probability of the new product being introduced is a sum of scenarios where each group wins and the product is launched based on their respective conditional probabilities.
• The calculation is expressed as: \[ P(\text{new product}) = (P(\text{new product} \mid A) \cdot P(A)) + (P(\text{new product} \mid B) \cdot P(B)) + (P(\text{new product} \mid C) \cdot P(C)) \]
• Each term of this equation involves a conditional probability multiplied by the probability of the associated condition happening, signifying a comprehensive view of all scenarios.

This technique shows how different probabilities combine to influence the overall likelihood of an event, making it vital for problems involving multiple influencing factors.

Mathematical Problem Solving

Mathematical problem solving is the art and science of identifying the necessary steps to arrive at a solution. In practical terms, it involves:

• Recognizing the given data and conditions, as in step 1, where the problem provides various probabilities of election outcomes and conditional scenarios.
• Applying the suitable mathematical theory or rule, such as using the Law of Total Probability in step 2, to bridge conditional probabilities with the overall event probability.
• Substituting values and computing, seen in step 3, where individual contributions are calculated (e.g., \(0.7 \times 0.5\) for group A).
• Summing these contributions as done in step 4 to arrive at the final probability \( P(\text{new product}) = 0.63 \).

By systematically addressing each aspect of the problem, from data recognition to final computation, students build a structured approach in mathematical problem solving, enhancing their analytical skills and proficiency in probability theory.