Problem 104 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A bag contains \(a\) white and \(b\) black balls. Two players \(A\) and \(B\) alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and and wins the game. If \(A\) begins the game and the probability of \(A\) winning the game is three times that of \(B\), then \(a: b=\) (A) \(2: 1\) (B) \(3: 1\) (C) \(3: 2\) (D) none of these

Step-by-Step Solution

Answer

The ratio \(a:b\) is \(3:2\), option (C).

Step 1: Identify Variables and Probabilities

Let the probability of drawing a white ball be \(P(W) = \frac{a}{a+b}\), and the probability of drawing a black ball be \(P(B) = \frac{b}{a+b}\). Player A wins if they draw a white ball on their turn, or if both draw black balls until A draws a white ball on a subsequent turn.

Step 2: Write the Expression for A Winning

The probability that A wins immediately is \(P(W)\). If A draws a black ball, B must also draw a black ball (probability \(P(B)\)), and then A must draw a white ball again to win, leading to the expression: \[ P_A = P(W) + P(B) \times P(B) \times P_A \]

Step 3: Solve for A's Probability

Solving the equation from Step 2: \[ P_A = \frac{a}{a+b} + \left(\frac{b}{a+b}\right)^2 \times P_A \]Simplifying gives: \[ P_A \left(1 - \left(\frac{b}{a+b}\right)^2\right) = \frac{a}{a+b}\]\[ P_A \left(\frac{a^2 + 2ab}{(a+b)^2}\right) = \frac{a}{a+b}\]\[ P_A = \frac{a(a+b)}{a^2 + 2ab}\]

Step 4: Write the Expression for B Winning

Player B wins if both draw black balls initially, and then B draws a white ball. The probability of this sequence is: \[ P_B = \left(\frac{b}{a+b}\right) \times \left(\frac{b}{a+b}\right) \times \frac{a}{a+b} = \frac{b^2 a}{(a+b)^3}\]

Step 5: Set Up Relationship Between A's and B's Probability

According to the problem, \(P_A = 3P_B\). Replace these probabilities with their simplified expressions: \[ \frac{a(a+b)}{a^2 + 2ab} = 3 \times \frac{b^2 a}{(a+b)^3}\]

Step 6: Simplify the Equation and Solve for a:b

Cantellation of common terms gives: \[ \frac{1}{a^2 + 2ab} = 3 \times \frac{b^2}{(a+b)^2}\]Further simplification leads to: \[ (a+b)^2 = 3b^2(a^2 + 2ab)\]Solving for \(\frac{a}{b}\) yields \(\frac{a}{b} = 3:2\).

Step 7: Match Final Ratio to Answer Choices

Comparing our calculated ratio \(3:2\) to the given answer choices, the correct answer is (C) \(3:2\).

Key Concepts

Ratio

A ratio helps us to compare two quantities, telling us how much of one exists in relation to the other. Here, when calculating the probability of winning a game by drawing balls from a bag, determining the right ratio of white to black balls is crucial.
In the exercise given, we identify that the probability of drawing a white ball relative to all balls in the bag is central. The ratio of white balls \( a \) to black balls \( b \), \( a:b \), is essential. It's not just about counting; it's about relation.

• Ratios simplify comparisons between two quantities.
• A ratio of \( 3:2 \), for instance, means for every 3 white balls, there are 2 black balls.

This ratio impacts the probability calculations by altering the likelihood of drawing either a white or a black ball in each turn. Understanding and managing these ratios allows us to solve more complex problems effectively.

Probability Theory

Probability theory is the branch of mathematics that deals with analyzing random phenomena. In this context, it helps us understand the chances of a player winning the game through drawing balls from a bag.
Probability provides a way to measure the likelihood of events using simple calculations. Here, the probability of player A winning, \( P_A \), and player B winning, \( P_B \), were calculated using probabilities of drawing specific ball colors:

• \( P(W) = \frac{a}{a+b} \): Probability of drawing a white ball
• \( P(B) = \frac{b}{a+b} \): Probability of drawing a black ball

Using probability theory, we build scenarios, such as a player drawing a black ball twice, which brings us a step closer to verify who might win the game. This systematic approach helps in deriving meaningful conclusions beyond mere guesswork.

Combinatorics

Combinatorics is the mathematics of counting and arrangement. It is closely related to probability since it often underpins the calculations determining the likelihood of an event.
In this exercise, combinatorics helps in calculating the various outcomes possible when players draw balls in turns. The combinations of sequences where player A or player B could draw balls until one wins are vast.
By employing combinatorial methods, we can figure out the probability dependencies, like if one must draw a series of black balls before drawing a white ball. Though often complex, combinatorics make these analyses feasible by breaking them into smaller, manageable calculations:

• Counting paths of sequences in drawing balls
• Breaking down multi-step probabilities

In applying combinatorics, understanding these permutations and combinations allows for a clearer pathway to solve the problem effectively.