Winning Tennis with Probability 0.3 and 0.7

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In summary, the probability of winning a game of tennis starting from even score Deuce (40-40) with a 0.3 chance of winning each point is 15.51%. The states are Deuce, Ad-In, Ad-Out, and Game, with Game being an absorbing state.
  • #1
WMDhamnekar
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Hi,

What is your probability of winning a game of tennis, starting from the even score Deuce (40-40), if your probability of winning each point is 0.3 and your opponent’s is 0.7?

1645684239650.png


My answer:
I think the sequence of independent trials are required to win a game of tennis starting from even score Deuce(40-40),each of which is a success with probability (0.3 × 0.3 =0.09) or a failure with probability (0.7 × 0.3 + 0.7 × 0.3 =0.42). Suppose, the independent trials to win a game of tennis are n. That means after (n-1) trials of failures, the nth trial is success.

= 15.51 %
 
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  • #2
Have you worked out the states and the state transition probability matrix?
I see the states as
Deuce, Ad-In, Ad-Out, Game

with Game being an absorbing state.
 
  • #3
Write out all the conditional probabilites P(W|X) where X = A, B, D. You will have a linear system of three unknowns and three equations, which can be solved by standard methods.

For example: If you are in state B, you must win the next point to have any chance of winning. If you lose the point you lose the game, so
P(W|B) = q P(W|D)
because if you do win the point (which you do with probability q), then you will be in state D and your probability of winning will be P(W|D).
 
  • #4
There is a useful shortcut in considering two point played at a time. If the probability of winning a point is , then the probability of winning the game () satisfies:
HenceWith , this gives .
 
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  • #5
PeroK said:
There is a useful shortcut in considering two point played at a time. If the probability of winning a point is , then the probability of winning the game () satisfies:
HenceWith , this gives .
Other way of seeing this: Probability of winning in two points is . Probability of losing in two points is . Probability of winning must therefore be

as otherwise we return to the same state.
 
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FAQ: Winning Tennis with Probability 0.3 and 0.7

What does a probability of 0.3 and 0.7 represent in tennis?

A probability of 0.3 represents the chance of one player winning a match, while a probability of 0.7 represents the chance of the other player winning. In this context, if Player A has a probability of 0.3, it means Player A is expected to win 30% of the time, while Player B, with a probability of 0.7, is expected to win 70% of the time.

How can these probabilities be used to predict match outcomes?

These probabilities can be used to predict match outcomes by applying them to a series of matches or simulations. By running simulations based on these probabilities, one can estimate the expected number of wins for each player over multiple matches, helping to understand their relative strengths.

What factors can influence the probabilities of winning in tennis?

Factors that can influence the probabilities of winning in tennis include player skill level, physical condition, playing style, surface type (e.g., clay, grass, hard court), weather conditions, and psychological factors such as confidence and experience in high-pressure situations.

Can these probabilities change during a match?

Yes, these probabilities can change during a match based on the players' performance, momentum shifts, injuries, and other dynamic factors. For example, if a player with a 0.3 probability starts winning games convincingly, their probability of winning may increase as the match progresses.

How can I calculate the expected number of wins for each player over a series of matches?

The expected number of wins can be calculated by multiplying the probability of winning by the total number of matches. For example, if Player A has a probability of 0.3 and plays 10 matches, the expected number of wins for Player A would be 0.3 * 10 = 3 wins, while Player B would be expected to win 7 matches (0.7 * 10 = 7 wins).

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