Question

In a kabaddi league, two matches are being played between Jaipur and Delhi. It is assumed that the outcomes of the two games are independent. The probability of Jaipur winning, drawing and losing the game against Delhi are \( \frac{1}{2} \), \( \frac{3}{10} \), and \( \frac{1}{5} \) respectively. Each team gets 5 points for win, 3 points for draw and 0 points for loss in a game. After two games, find the probability that Jaipur has more points than Delhi.

• A. \( \frac{1}{4} \)
• B. \( \frac{3}{20} \)
• C. \( \frac{11}{20} \)
• D. \( \frac{2}{5} \)

Hint:

When solving probability problems involving independent events, remember to multiply the probabilities of the individual events for each scenario. Then, sum the probabilities of favorable outcomes.

Answer 1

The Correct Option is C

Step 1: Define the possible outcomes for each match.
The possible outcomes for Jaipur's game against Delhi are:
- Win: Probability \( P(\text{Win}) = \frac{1}{2} \), Jaipur gets 5 points.
- Draw: Probability \( P(\text{Draw}) = \frac{3}{10} \), Jaipur gets 3 points.
- Lose: Probability \( P(\text{Lose}) = \frac{1}{5} \), Jaipur gets 0 points.
For each game, the points for Delhi will be the inverse of Jaipur's outcome:
- Win: Delhi gets 0 points.
- Draw: Delhi gets 3 points.
- Lose: Delhi gets 5 points.

Step 2: Find the total points for Jaipur and Delhi for all possible outcomes.
We can calculate the total points after two games by considering all possible outcomes of the two games. The four possibilities are:
1. Jaipur wins both games:
Jaipur gets \( 5 + 5 = 10 \) points, Delhi gets \( 0 + 0 = 0 \) points.
2. Jaipur wins the first game, draws the second game:
Jaipur gets \( 5 + 3 = 8 \) points, Delhi gets \( 0 + 3 = 3 \) points.
3. Jaipur wins the first game, loses the second game:
Jaipur gets \( 5 + 0 = 5 \) points, Delhi gets \( 0 + 5 = 5 \) points.
4. Jaipur draws both games:
Jaipur gets \( 3 + 3 = 6 \) points, Delhi gets \( 3 + 3 = 6 \) points.
5. Jaipur draws the first game, wins the second game:
Jaipur gets \( 3 + 5 = 8 \) points, Delhi gets \( 3 + 0 = 3 \) points.
6. Jaipur draws the first game, loses the second game:
Jaipur gets \( 3 + 0 = 3 \) points, Delhi gets \( 3 + 5 = 8 \) points.
7. Jaipur loses both games:
Jaipur gets \( 0 + 0 = 0 \) points, Delhi gets \( 5 + 5 = 10 \) points.
8. Jaipur loses the first game, wins the second game:
Jaipur gets \( 0 + 5 = 5 \) points, Delhi gets \( 5 + 0 = 5 \) points.

Step 3: Identify favorable outcomes for Jaipur having more points than Delhi.
From the list of possible outcomes above, Jaipur has more points than Delhi in the following cases:
- Jaipur wins both games (10 points vs 0 points)
- Jaipur wins the first game, draws the second game (8 points vs 3 points)
- Jaipur draws the first game, wins the second game (8 points vs 3 points)

Step 4: Calculate the probabilities of favorable outcomes.
- The probability of Jaipur winning both games:
\[ P(\text{Win, Win}) = P(\text{Win}) \times P(\text{Win}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
- The probability of Jaipur winning the first game and drawing the second game:
\[ P(\text{Win, Draw}) = P(\text{Win}) \times P(\text{Draw}) = \frac{1}{2} \times \frac{3}{10} = \frac{3}{20} \]
- The probability of Jaipur drawing the first game and winning the second game:
\[ P(\text{Draw, Win}) = P(\text{Draw}) \times P(\text{Win}) = \frac{3}{10} \times \frac{1}{2} = \frac{3}{20} \]
Thus, the total probability of Jaipur having more points than Delhi is:
\[ P(\text{Jaipur more points}) = \frac{1}{4} + \frac{3}{20} + \frac{3}{20} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20} \]

Step 5: Conclusion.
The probability that Jaipur has more points than Delhi is \( \frac{11}{20} \), so the correct answer is option (C).