Getting series outcome probabilities based on individual game probabilities

I have the probabilities for Team X beating Team Y in a game. Each team has two separate probabilities - the probability of winning at home and the probability of winning away.

These probabilities can then be used to calculate the expected winner of a best-of-7 series between the two team. Based on whether or not a team finished higher in the standings, they'll either play a maximum of 4 games at home, or a maximum 3. The format for these games is 2-2-1-1-1, with the higher seed playing twice at home, twice on the road, once at home (if necessary), once on the road (if necessary), and once at home (if necessary).

How could I find the probability that team one wins the series (wins 4 games) in X number (ranging from 4 to 7) of games? My initial thought was to multiply binomial probabilites, like so in R:

 games.out <- function(ph, pa) {
df <- data.frame(w = NA, x = NA, y = NA, z = NA)
for(games.played in 4:7){
home.games <- ceiling(games.played/2)
away.games <- floor(games.played/2)
home.wins <- 4 - away.games
away.wins <- away.games
loops <- games.played - 4
win.prob <- (dbinom(home.wins, home.games, ph)*dbinom(away.wins, away.games, pa))
home.wins <- home.wins + 1
away.wins <- away.wins - 1
for(i in 1:loops) {
temp.prob <- (dbinom(home.wins, home.games, ph)*dbinom(away.wins, away.games, pa))
home.wins <- home.wins + 1
away.wins <- away.wins -1
win.prob <- temp.prob + win.prob
}
df[1,games.played-3] <- win.prob
colnames(df)[games.played-3] <- paste0("Win", games.played)
}
return(df)
}


But, when summed, this code gave values that were much too large to make sense.

> games.out(.5, .5)
HW4     HW5      HW6       HW7
1 0.0625 0.15625 0.234375 0.2734375
> sum(games.out(.5, .5))
[1] 0.7265625


A similar question was asked here (How to calculate probability of winning best of 7 series), but that problem simply deals with one probability associated with all seven games - here, I'm looking at two different probabilities associated with potentially different numbers of games.

Any help is definitely appreciated - I've been banging my head against the wall for a couple of hours on this.

• They are too large because you expect the answer to be 0.5, in your example. The reason why they are too large is as follows. What you have calculated for HWN, where N is the number of games before the home team wins the series, is the value 0.5^N * choose(N, 4). This overestimates the probability. Instead, you should calculate 0.5^N * (choose(N, 4) - choose(N-1, 4)). This is because the number of games where the home team wins 4 games in N games includes the winning pathways where the home team wins 4 games in N-1 games. E.g. if N = 5 then the sequence WWWWL is counted...
– Alex
Apr 10, 2018 at 1:54
• ... when you use choose(5,4). You need to exclude these games that terminate early.
– Alex
Apr 10, 2018 at 1:56
• I was working towards something like this, but remember coming across a problem because there are two distinct probabilities. Since P(Home) and P(Away) are going to have different values, the probability of winning the series wouldn't be 0.5^N * (choose(N,4) - choose(N-1, 4)). It would be the sum of two formulas involving P(Home) and P(Away), correct? For example, if P(Home) = .65 and P(Away) = .5, the results would be different. Apr 10, 2018 at 2:56
• Yes, I am merely commenting on your calculations where both home and away win probs are 0.5. I have not looked at your code but it seemed to me that you can calculate the answer elegantly for this specific case. As of yet I do not know of an elegant way of calculating your desired result.
– Alex
Apr 10, 2018 at 4:02

Essentially you want to have a formula for the probability mass function of a negative binomial, except instead of observing the outcomes of identically distributed Bernoulli trials, we now have sequences of Bernoulli trials that are not identically distributed.

Unfortunately I do not know of a nice formula. However, we can derive a solution combinatorially by enumerating all possible paths that end when four team 1 wins or team 2 wins have been observed (these are respectively coded with 1's and 0's in the following functions). I will go through the steps by commenting in-line within these two functions in R.

get_succ_prob <- function(p, q) { # modify as necessary depending on pattern of home/away games
# probability team 1 wins at the i-th game
# can extend to other game win probs
c(p, p, q, q, p, q, p)
}


The main work is done by this function. n_trials is the number of games in the series, n_success is the number of games required to win the series, and succ_prob is the vector (generated by the previous function), such that the i-th element is the probability that team 1 wins the i-th game.

get_win_prob <- function(n_trials, n_success, succ_prob) {

stopifnot(length(succ_prob) == n_trials)

# enumerate 2^n_trials sequences of 0, 1.
tmp_mat <- as.matrix(do.call(expand.grid, rep(list(0:1), n_trials)))

# now terminate series of games at the first instance
# four zeroes or ones are observed
tmp_mat <- t(apply(tmp_mat, 1, function(x) {

threshold_to_NA <- suppressWarnings(min(min(which(cumsum(x == 0) == n_success)), min(which(cumsum(x == 1) == n_success))))
x[seq_len(n_trials) >= (threshold_to_NA + 1)] <- NA

return(x)
}
))

# Find unique paths to a series ending
tmp_mat <- unique(tmp_mat)

# calculate probability of observing series of game outcomes
prob_row <-
apply(tmp_mat, 1, function(x) prod(succ_prob * x + (1 - succ_prob) * (1 - x),
na.rm = T))

# make data frame and aggregate by team/series length
tmp_df <- data.frame(tmp_mat)

tmp_df$prob <- prob_row tmp_df$winning_team <-
2 - (rowSums(tmp_df[, seq_len(n_trials)], na.rm = T) == n_success)

tmp_df$trials_required <- n_trials - rowSums(is.na(tmp_df[, seq_len(n_trials)])) out_df <- aggregate(tmp_df$prob, by = list(tmp_df$trials_required, tmp_df$winning_team), FUN = sum)
names(out_df) <- c('games_required_to_win', 'winning_team', 'win_prob')

return(out_df)
}


> get_win_prob(7, 4, get_succ_prob(0.5, 0.5))
games_required_to_win winning_team win_prob
1                     4            1  0.06250
2                     5            1  0.12500
3                     6            1  0.15625
4                     7            1  0.15625
5                     4            2  0.06250
6                     5            2  0.12500
7                     6            2  0.15625
8                     7            2  0.15625


On more general home/away win probs:

> get_win_prob(7, 4, get_succ_prob(0.6, 0.3))
games_required_to_win winning_team win_prob
1                     4            1 0.032400
2                     5            1 0.116640
3                     6            1 0.093960
4                     7            1 0.192456
5                     4            2 0.078400
6                     5            2 0.120960
7                     6            2 0.236880
8                     7            2 0.128304