Best of Seven Outcome Probability Dec 2020

Best of Seven -- Calculate Outcome Probabilities

We have all watched sport series based on winning the best of seven — or first team to win four games.

I consider two teams: Team A and Team B. Given the probability of Team A winning any one game, $p(A)$ ; therefore, Team B has $p(B) = 1 - p(A)$ probability of winning any one game.

The two teams have eight possible outcomes:

4-3, 4-2, 4-1, 4-0,

0-4, 1-4, 2-4, 3-4

Using hypergeometric probability, I calculate the likelihood of each of the 8 outcomes given each Team A probability (0.0 to 1.0).

Formula

The formula is as follows for a 7 game series:

n = specific total number of games played for a given winning outcome ( $n = 6$ total for a 4-2 series)

$\left[\frac{(n-1)!}{3!(n-4)!}\right]p^4(1-p)^{n-4}$

Permutations

The expression $\frac{(n-1)!}{3!(n-4)!}$ calculates the number of permutations associated with a specific winning outcome (for example, 10 permutations for a 4-2 series where $n = 6$ ).

$\begin{align*} \\ \texttt{LLWWWW}\\ \texttt{WLLWWW}\\ \texttt{WWLLWW}\\ \texttt{WWWLLW}\\ \\ \texttt{LWLWWW}\\ \texttt{WLWLWW}\\ \texttt{WWLWLW}\\ \\ \texttt{LWWLWW}\\ \texttt{WLWWLW}\\ \\ \texttt{LWWWLW} \end{align*}$

Example

Given:

$\begin{align*} \\ p = 0.81,\\ n = 7,\\ \textrm{and }1-p = 0.19\\ \end{align*}$

Therefore, the probability of a 4-3 win is:

$\begin{align*} \\ \frac{(7-1)!}{3!(7-4)!}(0.81)^4(1-0.81)^{7-4}\\ \frac{6!}{3!3!}(0.81)^4(0.19)^3\\[4pt] (20)(0.00295)\\ 0.0591 \end{align*}$

Best of Seven Live Chart

The live chart looks best on a laptop or desktop screen. If viewing on a mobile device select ‘Request Desktop Website’ and use landscape orientation of the device.

Static Plots

Using PLOTLY live plots

Each chart series can be toggled off and on

by clicking its legend entry.

On a desktop device, reset the chart to full scale

by reloading the web page.

On a mobile device, reset the chart to full scale

by double tapping.

Best of Seven -- Winning Outcome Probabilities

Best of Seven Winning Outcomes Live Chart

This chart too looks best on a laptop or desktop screen. If viewing on a mobile device select ‘Request Desktop Website’ and use landscape orientation of the device.

Team A Winning Outcomes

The best of seven series steers the odds of winning in favor of the winning team. See table and chart below.

Game p(A)	Series p(A)
0.50	0.50
0.60	0.71
0.70	0.874
0.80	0.967
0.90	0.997
1.00	1.00

Best of Seven -- Winning Outcome Probabilities

Best of Seven Winning Outcomes (tall format for easier reading of graph) Live Chart

This chart too looks best on a laptop or desktop screen. If viewing on a mobile device select ‘Request Desktop Website’.

Reference

Best of Seven Jul 2022

Best of Seven Outcome Probability Dec 2020

Leave a Reply Cancel reply

Recent Posts

Archives

QuickLinks