Let us say that team A is playing team B in a best-of-7 playoff series. Let us also say that team A has probability p1 of winning Game 1 vs. team B, probability p2 of winning Game 2 vs. team B, probability p3 of winning Game 3 vs. team B, probability p4 of winning Game 4 vs. team B, probability p5 of winning Game 5 vs. team B, probability p6 of winning Game 6 vs. team B, and probability p7 of winning Game 7 vs. team B in that series. Then the probability of team A winning this best-of-7 series is pWLseries, where W is the number of games team A has won in this series, and L is the number of games team A has lost in this series. Since both W and L lie between 0 and 3, inclusive, there are 16 different states for an active best-of-7 playoff series: 00, 01, 02, 03, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, and 33. Here, then, in terms of the game victory probabilities defined above (and using the notation found in the Mathematica® software package), are the corresponding series victory probabilities for team A:

Best-of-7 series tied 3-games-all: Team A series victory probability = p33series[p7_] := p7

Best-of-7 series with team A trailing 3-games-2: Team A series victory probability = p23series[p6_, p7_] := p6*p7

Best-of-7 series with team A leading 3-games-2: Team A series victory probability = p32series[p6_, p7_] := p6 + p7 - p6*p7

Best-of-7 series with team A trailing 3-games-1: Team A series victory probability = p13series[p5_, p6_, p7_] := p5*p6*p7

Best-of-7 series tied 2-games-all: Team A series victory probability = p22series[p5_, p6_, p7_] := p5*p6 + p5*p7 + p6*p7 - 2*p5*p6*p7

Best-of-7 series with team A leading 3-games-1: Team A series victory probability = p31series[p5_, p6_, p7_] := p5 + p6 - p5*p6 + p7 - p5*p7 - p6*p7 + p5*p6*p7

Best-of-7 series with team A trailing 3-games-nil: Team A series victory probability = p03series[p4_, p5_, p6_, p7_] := p4*p5*p6*p7

Best-of-7 series with team A trailing 2-games-1: Team A series victory probability = p12series[p4_, p5_, p6_, p7_] := p4*p5*p6 + p4*p5*p7 + p4*p6*p7 + p5*p6*p7 - 3*p4*p5*p6*p7

Best-of-7 series with team A leading 2-games-1: Team A series victory probability = p21series[p4_, p5_, p6_, p7_] := p4*p5 + p4*p6 + p5*p6 - 2*p4*p5*p6 + p4*p7 + p5*p7 - 2*p4*p5*p7 + p6*p7 - 2*p4*p6*p7 - 2*p5*p6*p7 + 3*p4*p5*p6*p7

Best-of-7 series with team A leading 3-games-nil: Team A series victory probability = p30series[p4_, p5_, p6_, p7_] := p4 + p5 - p4*p5 + p6 - p4*p6 - p5*p6 + p4*p5*p6 + p7 - p4*p7 - p5*p7 + p4*p5*p7 - p6*p7 + p4*p6*p7 + p5*p6*p7 - p4*p5*p6*p7

Best-of-7 series with team A trailing 2-games-nil: Team A series victory probability = p02series[p3_, p4_, p5_, p6_, p7_] := p3*p4*p5*p6 + p3*p4*p5*p7 + p3*p4*p6*p7 + p3*p5*p6*p7 + p4*p5*p6*p7 - 4*p3*p4*p5*p6*p7

Best-of-7 series tied 1-game-all: Team A series victory probability = p11series[p3_, p4_, p5_, p6_, p7_] := p3*p4*p5 + p3*p4*p6 + p3*p5*p6 + p4*p5*p6 - 3*p3*p4*p5*p6 + p3*p4*p7 + p3*p5*p7 + p4*p5*p7 - 3*p3*p4*p5*p7 + p3*p6*p7 + p4*p6*p7 - 3*p3*p4*p6*p7 + p5*p6*p7 - 3*p3*p5*p6*p7 - 3*p4*p5*p6*p7 + 6*p3*p4*p5*p6*p7

Best-of-7 series with team A leading 2-games-nil: Team A series victory probability = p20series[p3_, p4_, p5_, p6_, p7_] := p3*p4 + p3*p5 + p4*p5 - 2*p3*p4*p5 + p3*p6 + p4*p6 - 2*p3*p4*p6 + p5*p6 - 2*p3*p5*p6 - 2*p4*p5*p6 + 3*p3*p4*p5*p6 + p3*p7 + p4*p7 - 2*p3*p4*p7 + p5*p7 - 2*p3*p5*p7 - 2*p4*p5*p7 + 3*p3*p4*p5*p7 + p6*p7 - 2*p3*p6*p7 - 2*p4*p6*p7 + 3*p3*p4*p6*p7 - 2*p5*p6*p7 + 3*p3*p5*p6*p7 + 3*p4*p5*p6*p7 - 4*p3*p4*p5*p6*p7

Best-of-7 series with team A trailing 1-game-nil: Team A series victory probability = p01series[p2_, p3_, p4_, p5_, p6_, p7_] := p2*p3*p4*p5 + p2*p3*p4*p6 + p2*p3*p5*p6 + p2*p4*p5*p6 + p3*p4*p5*p6 - 4*p2*p3*p4*p5*p6 + p2*p3*p4*p7 + p2*p3*p5*p7 + p2*p4*p5*p7 + p3*p4*p5*p7 - 4*p2*p3*p4*p5*p7 + p2*p3*p6*p7 + p2*p4*p6*p7 + p3*p4*p6*p7 - 4*p2*p3*p4*p6*p7 + p2*p5*p6*p7 + p3*p5*p6*p7 - 4*p2*p3*p5*p6*p7 + p4*p5*p6*p7 - 4*p2*p4*p5*p6*p7 - 4*p3*p4*p5*p6*p7 + 10*p2*p3*p4*p5*p6*p7

Best-of-7 series with team A leading 1-game-nil: Team A series victory probability = p10series[p2_, p3_, p4_, p5_, p6_, p7_] := p2*p3*p4 + p2*p3*p5 + p2*p4*p5 + p3*p4*p5 - 3*p2*p3*p4*p5 + p2*p3*p6 + p2*p4*p6 + p3*p4*p6 - 3*p2*p3*p4*p6 + p2*p5*p6 + p3*p5*p6 - 3*p2*p3*p5*p6 + p4*p5*p6 - 3*p2*p4*p5*p6 - 3*p3*p4*p5*p6 + 6*p2*p3*p4*p5*p6 + p2*p3*p7 + p2*p4*p7 + p3*p4*p7 - 3*p2*p3*p4*p7 + p2*p5*p7 + p3*p5*p7 - 3*p2*p3*p5*p7 + p4*p5*p7 - 3*p2*p4*p5*p7 - 3*p3*p4*p5*p7 + 6*p2*p3*p4*p5*p7 + p2*p6*p7 + p3*p6*p7 - 3*p2*p3*p6*p7 + p4*p6*p7 - 3*p2*p4*p6*p7 - 3*p3*p4*p6*p7 + 6*p2*p3*p4*p6*p7 + p5*p6*p7 - 3*p2*p5*p6*p7 - 3*p3*p5*p6*p7 + 6*p2*p3*p5*p6*p7 - 3*p4*p5*p6*p7 + 6*p2*p4*p5*p6*p7 + 6*p3*p4*p5*p6*p7 - 10*p2*p3*p4*p5*p6*p7

Best-of-7 series tied nil-games-all: Team A series victory probability = p00series[p1_, p2_, p3_, p4_, p5_, p6_, p7_] := p1*p2*p3*p4 + p1*p2*p3*p5 + p1*p2*p4*p5 + p1*p3*p4*p5 + p2*p3*p4*p5 - 4*p1*p2*p3*p4*p5 + p1*p2*p3*p6 + p1*p2*p4*p6 + p1*p3*p4*p6 + p2*p3*p4*p6 - 4*p1*p2*p3*p4*p6 + p1*p2*p5*p6 + p1*p3*p5*p6 + p2*p3*p5*p6 - 4*p1*p2*p3*p5*p6 + p1*p4*p5*p6 + p2*p4*p5*p6 - 4*p1*p2*p4*p5*p6 + p3*p4*p5*p6 - 4*p1*p3*p4*p5*p6 - 4*p2*p3*p4*p5*p6 + 10*p1*p2*p3*p4*p5*p6 + p1*p2*p3*p7 + p1*p2*p4*p7 + p1*p3*p4*p7 + p2*p3*p4*p7 - 4*p1*p2*p3*p4*p7 + p1*p2*p5*p7 + p1*p3*p5*p7 + p2*p3*p5*p7 - 4*p1*p2*p3*p5*p7 + p1*p4*p5*p7 + p2*p4*p5*p7 - 4*p1*p2*p4*p5*p7 + p3*p4*p5*p7 - 4*p1*p3*p4*p5*p7 - 4*p2*p3*p4*p5*p7 + 10*p1*p2*p3*p4*p5*p7 + p1*p2*p6*p7 + p1*p3*p6*p7 + p2*p3*p6*p7 - 4*p1*p2*p3*p6*p7 + p1*p4*p6*p7 + p2*p4*p6*p7 - 4*p1*p2*p4*p6*p7 + p3*p4*p6*p7 - 4*p1*p3*p4*p6*p7 - 4*p2*p3*p4*p6*p7 + 10*p1*p2*p3*p4*p6*p7 + p1*p5*p6*p7 + p2*p5*p6*p7 - 4*p1*p2*p5*p6*p7 + p3*p5*p6*p7 - 4*p1*p3*p5*p6*p7 - 4*p2*p3*p5*p6*p7 + 10*p1*p2*p3*p5*p6*p7 + p4*p5*p6*p7 - 4*p1*p4*p5*p6*p7 - 4*p2*p4*p5*p6*p7 + 10*p1*p2*p4*p5*p6*p7 - 4*p3*p4*p5*p6*p7 + 10*p1*p3*p4*p5*p6*p7 + 10*p2*p3*p4*p5*p6*p7 - 20*p1*p2*p3*p4*p5*p6*p7

Example problem: Let us assume that team A holds the home-court/field/ice advantage, and thus will play Games 1, 2, 5, and 7 at home. If team A has a historical victory probability of pH vs. team B in games played at home, and of pV vs. team B in games played on the road, then what is the team A pre-series victory probability? Solution: Using p00series[pH, pH, pV, pV, pH, pV, pH] in Mathematica®, we find pH^4 + 12*pH^3*pV - 12*pH^4*pV + 18*pH^2*pV^2 - 48*pH^3*pV^2 + 30*pH^4*pV^2 + 4*pH*pV^3 - 24*pH^2*pV^3 + 40*pH^3*pV^3 - 20*pH^4*pV^3. Note that the same resulting formula falls out if the home games are 1, 2, 6, and 7 (as in a 2-3-2 series) instead of 1, 2, 5, and 7 (as in a 2-2-1-1-1 series).