Problem 118 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(A\) and \(B\) play a game of tennis. The situation of the game is as follows: if one scores two consecutive points after a deuce, he wins. If loss of a point is followed by win of a point, it is deuce. The probability of a server to win a point is \(\frac{2}{3}\). The game is at deuce and \(A\) is serving. Probability that \(A\) will win the match is (serves are changed after each game ) (A) \(\frac{1}{4}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{2}\) (D) none of these

Step-by-Step Solution

Answer

The probability that A will win the match is \( \frac{4}{5} \), which is (D) none of these.

Step 1: Calculate the Probability of Winning and Losing Points

Since A is serving, he has a probability of \( \frac{2}{3} \) to win a point and \( \frac{1}{3} \) to lose a point. To win the game, A must win two consecutive points.

Step 2: Calculate the Probability That A Wins Two Consecutive Points

The probability of A winning two consecutive points is given by the product of A winning two points consecutively: \( \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \frac{4}{9} \).

Step 3: Consider Scenarios Where The Game Returns to Deuce

If A wins the first point (probability \( \frac{2}{3} \)) and loses the second point (probability \( \frac{1}{3} \)), the game returns to deuce. The probability of this happening is \( \left( \frac{2}{3} \right) \times \left( \frac{1}{3} \right) = \frac{2}{9} \).

Step 4: Consider Scenarios Where A Loses the First Point and Then Wins to Return to Deuce

If A loses the first point (probability \( \frac{1}{3} \)) and wins the next point (probability \( \frac{2}{3} \)), it returns to deuce. The probability for this is \( \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \).

Step 5: Calculate the Total Probability of Returning to Deuce

The total probability of returning to deuce (either scenario from Step 3 or 4) is \( \frac{2}{9} + \frac{2}{9} = \frac{4}{9} \).

Step 6: Set Up the Equation for A Winning the Game

Using recursive probabilities: let the probability of A winning the game from deuce be \( P \). Then, \( P = \frac{4}{9} + \frac{4}{9} \cdot P \). We solve for \( P \).

Step 7: Solve the Equation for P

Rearrange the equation: \( P - \frac{4}{9} \cdot P = \frac{4}{9} \). Thus, \( P \left( 1 - \frac{4}{9} \right) = \frac{4}{9} \), which simplifies to \( \frac{5}{9}P = \frac{4}{9} \). Solving for \( P \), we find \( P = \frac{4}{5} \).

Step 8: Select the Correct Answer from the Options

Since we calculated the probability for A to win the match from deuce as \( \frac{4}{5} \), and since this is not among the given options, the answer is (D) none of these.

Key Concepts

Consecutive Points

In tennis, achieving a victory from a deuce situation requires understanding the concept of consecutive points. When players reach a deuce, it means both competitors have the same score, and a player must win two points in a row to secure the game victory. This is because the first point won gives a player the advantage, and a subsequent point converts that advantage into a win. If, however, the player loses a point after gaining an advantage, the score returns to deuce. In the exercise, player A needs to win two consecutive points after reaching a deuce to win the game. Given that player A has a probability of winning a point of \( \frac{2}{3} \), the probability for A to win two consecutive points is \( \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \frac{4}{9} \).

Deuce in Tennis

A deuce in tennis represents a pivotal point where the game is evenly matched. This occurs when both players have won an equal number of points, and no player has yet gained the advantage needed for victory. During a deuce, the game's dynamics change, as players need to win extra points to clinch the game. In this context, winning the game involves a tactical battle over these critical points. Two outcomes can arise from a deuce:

• The player wins two consecutive points to win the game, moving from deuce to advantage and then to game.
• The player strives to win an initial point to gain an advantage, then a second point to win, or fails, and returns to deuce.

In the illustrated exercise, player A, serving from deuce, not only needs concentration but a strategy to convert the situation into a win, reflecting the strategic nature of tennis at such crucial moments.

Recursive Probability

Recursive probability is a powerful tool in calculating the likelihood of complex sequences of events, such as those found in a tennis game that hits multiple deuces. In this scenario, each sequence leading back to deuce requires its probability calculation, influencing overall chances. In recursive scenarios, like our tennis match, probabilities build on each other cyclically. For example:

• If player A wins the first point and loses the second, returning to deuce occurs with a probability of \( \frac{2}{9} \).
• Likewise, the opposite sequence of losing then winning also results in returning to deuce with the same probability of \( \frac{2}{9} \).

The combined recursive probability of returning to deuce is therefore \( \frac{4}{9} \), impacting the overall probability of victory. Using these recursive probabilities, an equation was set up in the solution: \( P = \frac{4}{9} + \frac{4}{9} \cdot P \), where \( P \) is the probability of winning from deuce. This equation captures how often returning to deuce affects the probability of eventually winning, illustrating the elegance and utility of recursive probability theory in determining outcomes.