# Probability of a player winning a best of 7

• disregardthat
In summary, the conversation discusses the calculation of the probability of a player winning the first game and the entire match in a best of 7 series. The equation for this probability is given, and the conversation then delves into questions about interpreting this probability, calculating it for a random variable p, and the difference between calculating it for the entire match versus given that the player has won the first match. It is noted that the underlying assumption of a fixed p may not be accurate in sports matchups.
disregardthat
This discussion popped up on a different forum and I'd like to hear some opinions on this.

Suppose player A and B are playing each other in a best of 7 match. Player A has probability p of winning against B. We want to calculate the probability of a player winning the first game and the entire match.

The probability of this may be calculated as such: We calculate the probability of A winning the first game and the entire match. Then A wins the first and last game, and B wins either of k out of 2+k matches in between, for k = 0,1,2,3. Then the probability becomes

##\sum^3_{k=0} p^4(1-p)^k {2+k \choose k}##

Conversely, the same happening for B has probability:

##\sum^3_{k=0} p^k(1-p)^4 {2+k \choose k}##

Summing these yields

##2p^6 - 6p^5 + 5p^4 - p + 1##.

The observed probability of this happening is 0.73. On to the questions:

Solving ##2p^6 - 6p^5 + 5p^4 - p + 1 = 0.73## yields ##p \approx 0.3## or ##0.7##.

On to the questions:1) Can one interpret 0.7 as the expected probability of the stronger player winning the first game and the entire match? I'm not so sure.

2) Given p as a random variable, how can one calculate the probability of the stronger player winning the entire match and the first match?

3) Given p as a random variable, how can one calculate the probability of a player winning the entire match GIVEN he wins the first match?

4) What is the difference between 2) and 3), i.e. how can they be interpreted?

5) What can the entire data (list of observed results) say about the distribution of p, or any of the above?

I am sure there is a lot of confusion here. I'd appreciate someone setting this straight.

Last edited:
disregardthat said:
The observed probability of this happening is 0.73.

This rings a warning bell. From your setup, you are looking at the playoff games in one or more sports. Your underlying assumption is that p is fixed, but this is certainly not true and will depend on the actual teams. You are trying to infer a parameter which will change with the matchup from a large number of matchups.

disregardthat said:
Can one interpret 0.7 as the expected probability of the stronger player winning the first game and the entire match? I'm not so sure.
No, by your own definition, p is the probability of the stronger team (in a fixed matchup) winning any particular game.

disregardthat said:
Given p as a random variable, how can one calculate the probability of the stronger player winning the entire match and the first match?
A priori you could do this if you knew the distribution of p, which is going to depend on the possible matchups. Naturally, this probability is going to be larger for larger p (for p = 1 the probability is 1).

Thanks. But the expression, as written, does make sense for a stochastic variable p, right?

So we have an estimate of the new variable, ##X =2p^6 - 6p^5 + 5p^4 - p + 1##.

Let's say we know something about p (e.g. normally distributed, mean 0.5 and standard deviation s), what can we infer about X, and in particular P(X >= 0.73)?

If you see p as a stochastic variable, then the probability of having one team win the first game and the series is a stochastic variable and has a certain distribution. In order to find out the probability given this distribution you will have to integrate over the distribution of p. Inserting the expectation value of p into the formula will generally not give you the expectation value of X.

## 1. What is the probability of a player winning a best of 7 match?

The probability of a player winning a best of 7 match depends on several factors, including the skill level of both players, previous match history, and any external factors that may influence the outcome. It is not possible to determine a specific probability without considering these variables.

## 2. How does the probability of winning change throughout a best of 7 match?

The probability of winning typically changes throughout a best of 7 match as each player wins or loses individual games. For example, if a player wins the first three games, their probability of winning the overall match will increase. However, if their opponent wins the next three games, the probability will become more evenly split.

## 3. Is there a way to calculate the exact probability of winning a best of 7 match?

While there are methods for calculating probability in certain situations, such as coin flips or dice rolls, it is not possible to accurately calculate the exact probability of winning a best of 7 match due to the many variables involved. It is important to remember that probability is not an exact science and can only provide estimates.

## 4. Can the probability of winning a best of 7 match be influenced by outside factors?

Yes, the probability of winning a best of 7 match can be influenced by outside factors such as fatigue, injuries, or weather conditions. These factors may affect the performance of one or both players and can impact the overall outcome of the match.

## 5. How can understanding probability improve a player's chances of winning a best of 7 match?

Understanding probability can help a player make more strategic decisions during a best of 7 match. By considering the likelihood of certain outcomes, a player can adjust their gameplay and potentially increase their chances of winning. However, it is important to remember that probability is not a guarantee and there are always other factors at play in any competition.

Replies
1
Views
1K
Replies
7
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
75
Views
7K
Replies
4
Views
6K
Replies
9
Views
2K
Replies
1
Views
303
Replies
8
Views
2K
Replies
3
Views
1K