Problem 22 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A bag contains 9 white balls and 5 black balls. Another bag contains 8 white balls and 6 black balls. One ball is transferred from the first bag into the second, and then a ball is drawn from the latter. The probability that it will be a white ball is (a) \(1 / 2\) (b) \(1 / 5\) (c) \(1 / 21\) (d) \(121 / 210\)

Step-by-Step Solution

Answer

The probability that a drawn ball will be white is \( \frac{121}{210} \) (option d).

Step 1: Determine Initial Probabilities

Calculate the probability of transferring a white ball and a black ball from the first bag, and the consequences for the second bag.- Probability of transferring a white ball: \( \frac{9}{14} \)- Probability of transferring a black ball: \( \frac{5}{14} \)

Step 2: Calculate Probabilities After White Ball Transferred

If a white ball is transferred, the second bag has 9 white and 6 black balls.- Probability of drawing a white ball from the second bag: \( \frac{9}{15} = \frac{3}{5} \)

Step 3: Calculate Probabilities After Black Ball Transferred

If a black ball is transferred, the second bag has 8 white and 7 black balls.- Probability of drawing a white ball from the second bag: \( \frac{8}{15} \)

Step 4: Combine Results Using Total Probability

Calculate the total probability of drawing a white ball from the second bag using the law of total probability.\[ P(\text{White}) = \left( \frac{9}{14} \cdot \frac{3}{5} \right) + \left( \frac{5}{14} \cdot \frac{8}{15} \right) \]

Step 5: Simplify Final Probability

Simplify the expression to determine the final probability:\[P(\text{White}) = \frac{27}{70} + \frac{40}{210} = \frac{81 + 40}{210} = \frac{121}{210}\]

Key Concepts

Total Probability Theorem

The Total Probability Theorem is a fundamental principle in probability theory. It helps us determine the probability of an event based on several paths or scenarios. Imagine you have multiple ways to achieve the same outcome or event. Each path has a certain probability, and the theorem combines these probabilities.
This theorem is crucial when you're dealing with compound events. In our case, drawing a white ball from the second bag after a ball has been transferred from the first one is a compound event.
To apply the Total Probability Theorem, you break down the problem like this:

• Identify each possible scenario (e.g., transferring a white or black ball).
• Calculate the probability for each scenario separately.
• Combine the probabilities of these scenarios to get the total probability of the main event (drawing a white ball).

This step-by-step approach provides clarity and ensures that every possible outcome is considered.

Probability Theory

Probability Theory is the branch of mathematics that deals with the likelihood of different outcomes. It helps us understand and quantify uncertainty.
In probability, the outcome of an event is a numerical value between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means it will surely happen. Anything in between indicates varying levels of possibility.
The art of probability isn't just about numbers; it involves understanding how these numbers represent real-world situations. Like in this problem, where we have specific basic events:

• Transferring a white ball from the first bag to the second.
• Transferring a black ball instead.
• Drawing a white ball from the second bag afterward.

By assigning probabilities to these events, we can predict the likelihood of drawing a white ball. Understanding probability theory makes it easier to solve such problems by breaking them down into manageable parts.

Mathematical Problem Solving

Mathematical Problem Solving is a skill that involves using various mathematical concepts and logic to find solutions to problems. Each problem can often be broken down into smaller, more manageable parts.

In the given ball transfer problem, the steps walkthrough the process of determining which ball is likely to be drawn from the second bag. Mathematical problem solving here involves:

• Breaking down the problem: Identify what happens when a ball is transferred and how it affects the second bag.
• Utilizing known formulas: Use the Total Probability Theorem to piece together the result.
• Step-by-step logic: By taking small, carefully calculated steps, we reach a solution without jumping to conclusions.

By applying these problem-solving techniques, you enhance your mathematical intuition and ability to tackle complex problems efficiently. This structured approach ensures you consider all aspects of a problem and arrive at a sound conclusion. Each step becomes clearer, and errors can be minimized.