Mendelian Genetics worksheet illustrating the inheritance of fruit color in tomatoes using a Punnett square.

Problem Analysis:

The problem involves a scenario where two players, P1 and P2, are playing a game of tennis. The rules are as follows:

1. Game Scoring:
- A player must win at least 4 points to win the game.
- A player must win by at least 2 points over their opponent.

2. Set Scoring:
- A set is won by winning 6 games.
- If the score reaches 5-5 (or 6-6), a tiebreak is played to determine the winner of the set.

3. Tiebreak Rules:
- In a tiebreak, the first player to reach 7 points wins the set, provided they lead by at least 2 points.
- If the score reaches 6-6 in the tiebreak, the game continues until one player leads by 2 points.

4. Match Scoring:
- The match is won by winning 2 out of 3 sets.

The task is to calculate the probability that P1 wins the match given the following probabilities:
- Probability that P1 wins a point when serving: \( p \)
- Probability that P1 wins a point when receiving: \( q \)

Solution Approach:

To solve this problem, we need to break it down into smaller components and calculate the probabilities step by step. Here’s the structured approach:

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#### Step 1: Probability of Winning a Game
A game is won by winning at least 4 points with a margin of at least 2 points. We need to consider the different scenarios for winning a game based on whether P1 is serving or receiving.

##### Case 1: P1 Serving
Let \( p \) be the probability that P1 wins a point when serving. The possible outcomes for a game are:
- P1 wins 4 points before P2 wins 3 points.
- The game goes to deuce (4-4) and then P1 wins from there.

The probability of P1 winning a game while serving can be calculated using the binomial distribution and recursive formulas for deuce scenarios. However, for simplicity, we use the known formula for the probability of winning a game when serving:
\[
P(\text{P1 wins game | serving}) = \frac{p^4}{p^4 + 4pq^3 + 6p^2q^2(1-p-q)}
\]

##### Case 2: P1 Receiving
Let \( q \) be the probability that P1 wins a point when receiving. Similarly, the probability of P1 winning a game while receiving is:
\[
P(\text{P1 wins game | receiving}) = \frac{q^4}{q^4 + 4qp^3 + 6q^2p^2(1-p-q)}
\]

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#### Step 2: Probability of Winning a Set
A set is won by winning 6 games, with a tiebreak if the score reaches 5-5 or 6-6. We need to calculate the probability of P1 winning a set based on the probabilities of winning individual games.

##### General Case:
The probability of P1 winning a set can be calculated using the binomial distribution for winning 6 games out of 12 (since a tiebreak is required if the score reaches 5-5). However, the exact calculation involves summing over all possible ways P1 can win 6 games before P2 wins 5 games, considering the probabilities of winning each game.

For simplicity, let:
- \( G_s \) be the probability that P1 wins a game when serving.
- \( G_r \) be the probability that P1 wins a game when receiving.

The overall probability of P1 winning a set, \( P(\text{P1 wins set}) \), can be approximated using the recursive nature of set scoring and tiebreaks. This involves complex combinatorial calculations, but the general form is:
\[
P(\text{P1 wins set}) = \sum_{k=0}^{5} \binom{11}{k} G_s^k (1-G_s)^{11-k} \cdot \text{(probability of winning remaining games/tiebreak)}
\]

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#### Step 3: Probability of Winning the Match
The match is won by winning 2 out of 3 sets. Using the binomial distribution, the probability of P1 winning the match is:
\[
P(\text{P1 wins match}) = \sum_{k=2}^{3} \binom{3}{k} \left( P(\text{P1 wins set}) \right)^k \left( 1 - P(\text{P1 wins set}) \right)^{3-k}
\]

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Final Answer:

The exact probability of P1 winning the match depends on the values of \( p \) and \( q \). However, the general formula for the probability of P1 winning the match is:
\[
\boxed{P(\text{P1 wins match}) = \sum_{k=2}^{3} \binom{3}{k} \left( P(\text{P1 wins set}) \right)^k \left( 1 - P(\text{P1 wins set}) \right)^{3-k}}
\]
where \( P(\text{P1 wins set}) \) is determined by the probabilities of winning games and accounting for tiebreaks.