CODEWITHSUNDEEP
pythonprobability
user3047572
user3047572

Reputation: 49

Working out probability of a team win

I'm trying to work out the probability of team A winning a game of squash. Both teams have 3 members:

Team A: Have abilities(r) 40, 50 and 70 - Team B have abilities(r) 75, 25 and 30.

The team winning at least two games wins the match. If team A play in the order given above and teamB pick a random order:

(a) Estimate the probability of TeamA winning

(b) If the match ends as soon as one team has won two games, what is the expected number of games played.

I have used the equation to work out probability of team A winning one round: (Probability of Team A winning) = rA / (rA + rB)

So far I have just tried to calculate the chance of Team A winning.

import random

def game(a,b,c,d,e,f):

    overallprob = 0

    for item in [a, b, c]:
        probA = item / (item + d)
        overallprob = overallprob + probA

    for item in [a, b, c]:
        probA = item / (item + e)
        overallprob = overallprob + probA

    for item in [a, b, c]:
        probA = item / (item + f)
        overallprob = overallprob + probA

    print "Chances of team A winning =",round((overallprob / 9*100),2),"%"

game(40.0,50.0,60.0,75.0,25.0,30.0)

Which prints:

Chances of team A winning = 56.04 %

I am not sure if this is correct and I was wondering if I could get any help with part (b) as I'm not sure where to begin

Upvotes: 1

Views: 2733

Answers (2)

FMc
FMc

Reputation: 42421

from itertools import permutations, product

def main():
    teamA = [40, 50, 70]
    teamB = [75, 25, 30]

    # Compute two averages by processing every possible match:
    #   pa   Probability that Team A wins a match.
    #   ng   Expected N of games in a match.
    tot_pa, tot_ng, n = (0, 0, 0)
    for As, Bs in product(permutations(teamA), permutations(teamB)):
        pa, ng = prob_a_wins(As, Bs)
        tot_pa += pa
        tot_ng += ng
        n      += 1

    print tot_pa / n  # 0.61233
    print tot_ng / n  # 2.50580

def prob_a_wins(As, Bs):
    # Probabilities that Team A wins game 1, 2, 3, and the match.
    g1, g2, g3 = [ a / float(a + b) for a, b in zip(As, Bs) ]
    pa = (
        g1       * g2            +  # win g1 and g2
        g1       * (1 - g2) * g3 +  # win g1 and g3
        (1 - g1) * g2       * g3    # win g2 and g3
    )

    # Probabability of a two-game match, and expected N of games.
    two = (
        g1       * g2 +        # win  g1 and g2
        (1 - g1) * (1 - g2)    # lose g1 and g2
    )
    ng  = two * 2  +  (1 - two) * 3

    return (pa, ng)

main()

Upvotes: 1

Hugh Bothwell
Hugh Bothwell

Reputation: 56714

What you are calculating are the odds that a random player from team A will beat a random player from team B.

Once you take into account the possible pairings between teams, the odds that team A will beat a random ordering of team B becomes 61.23%

from itertools import permutations

def match_win_prob(ra, rb):
    """
    Given the skills of players a and b,
    return the probability of player a beating player b
    """
    return ra / float(ra + rb)

def team_win_prob(x, y, z):
    """
    Given the probability of a win in each match,
    return the probability of being first to two games
    (same as best-of-three)
    """
    return x*y + x*(1.-y)*z + (1.-x)*y*z

def game_win_prob(As, Bs):
    pairs = [[match_win_prob(a, b) for b in Bs] for a in As]
    ways, prob = 0, 0.
    for i,j,k in permutations(pairs):
        ways += 1
        prob += team_win_prob(i[0], j[1], k[2])
    return prob / ways

def main():
    p = game_win_prob([40, 50, 70], [75, 25, 30])
    print('Chances of team A winning = {:.2f} %'.format(p * 100.))

if __name__=="__main__":
    main()

Upvotes: 0

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