Badminton

[BMAD]

John amd Julie are both good badminton players.  Is it more likely that julie will win 5 out of 9 games or 4 out of 7?

[k-man]

Given that John and Julie play on the same level and have 50/50 chance of winning any particular game, I agree with Bonanova that Julie has 50% chance of winning whatever series they choose to play. However, I think the OP is asking about the probability of Julie winning with the score 4:3 in the 7 game series vs. the probability of winning 5:4 in the 9 game series. Those probabilities are different...

The probability that the outcome of a 9 game series will be 5:4 is 63/256, but the probability that the outcome of a 7 game series will be 4:3 is 35/128 and is higher, so Julie is more likely to win 4 games out of 7 than 5 games out of 9.

[bonanova]

One of them must win. Given no information on their relative skill, I would conclude they are equally likely to win. As to which score is the more likely score, it depends. If they just play games, until someone has achieved one of the winning situations, then 4 of 7 occurs first, and is the 100% most likely winning score. Neither of them could win 5 of 9. But if it's set up to be 5 of 9, so that winning 4 games is not enough, OR set up to be 4 of 7, where first to 4 wins, then it seems to be a tossup: she has a 50% win expectation either way - it would not be to her advantage for it to be 4/7 or 5/9.

[dgreening]

If we go with the obvious.

4/7 [57.1%] is slightly higher than 5/9 [55.5%].

If both are equally good, then the odds would suggest that it is more likely to win 5 out of 9 [than 4  out of 7]

 

[harey]

Scores:

0:0 (p=1), then
1:0 or 0:1 (p=0.5+0.5), then
2:0 or 1:1 or 0:2 (p=0.25+0.5+0.25)
...
7:0 6:1 5:2 4:3...

Pascal's triangle, but I will not hazard myself into calculations.

Edited August 24, 2015 by harey

[jasen]

Use This Code at microsoft word 2010 macro VBA to test this problem.

Sub BadmintonTest()
Dim JohnWin As Single
Dim JulieWin As Single
Dim P5of9 As Single
Dim P4of7 As Single
    Randomize
    For i = 1 To 500000
        For j = 1 To 9
            NewValue = Int((2 * Rnd) + 1)
            If NewValue = 1 Then JohnWin = JohnWin + 1
            If NewValue = 2 Then JulieWin = JulieWin + 1
            If JulieWin = 5 Then P5of9 = P5of9 + 1
        Next j
        JulieWin = 0
    Next i
    Selection.TypeText Text:=" percentage 5of9 = " + str(P5of9 / 5000000)

    For i = 1 To 500000
        For j = 1 To 7
            NewValue = Int((2 * Rnd) + 1)
            If NewValue = 1 Then JohnWin = JohnWin + 1
            If NewValue = 2 Then JulieWin = JulieWin + 1
            If JulieWin = 4 Then P4of7 = P4of7 + 1
        Next j
        JulieWin = 0
    Next i
    Selection.TypeText Text:="     percentage 4of7 = " + str(P4of7 /I run th 5000000)
End Sub

----------------------

I run the code above and find this result

percentage 5of9 =  .0754452      percentage 4of7 =  .0727584
percentage 5of9 =  .0753496      percentage 4of7 =  .072915 
percentage 5of9 =  .0752444      percentage 4of7 =  .0727264 
percentage 5of9 =  .0751622      percentage 4of7 =  .0725046

So it more likely that julie will win 5 out of 9 games than 4 out of 7

[jasen]

Sory I thought I was wrong, The code should be like this :

Sub BadmintonTest()
Dim JohnWin As Single
Dim JulieWin As Single
Dim P5of9 As Single
Dim P4of7 As Single
    Randomize
    For i = 1 To 500000
        For j = 1 To 9
            NewValue = Int((2 * Rnd) + 1)
            If NewValue = 1 Then JohnWin = JohnWin + 1
            If NewValue = 2 Then JulieWin = JulieWin + 1
        Next j
        If JulieWin = 5 Then P5of9 = P5of9 + 1
        JulieWin = 0
    Next i
    Selection.TypeText Text:=" percentage 5of9 = " + str(P5of9 / 5000000)

    For i = 1 To 500000
        For j = 1 To 7
            NewValue = Int((2 * Rnd) + 1)
            If NewValue = 1 Then JohnWin = JohnWin + 1
            If NewValue = 2 Then JulieWin = JulieWin + 1
        Next j
        If JulieWin = 4 Then P4of7 = P4of7 + 1
        JulieWin = 0
    Next i
    Selection.TypeText Text:="      percentage 4of7 = " + str(P4of7 / 5000000)
End Sub

---------------------

after 3 times I get this :
percentage 5of9 =  .0245682      percentage 4of7 =  .0274882
percentage 5of9 =  .0246054      percentage 4of7 =  .0273542
percentage 5of9 =  .0245194      percentage 4of7 =  .027356

So it more likely that julie will win 4 out of 7 games than 5 out of 9

 

[bonanova]

Yeah, my reasoning was wrong. One can say that matches are more likely to end with one of the scores. Julie's half of those matches are then more numerous than her half of the matches that end in the other score. Nice little puzzle.

[bubbled]

I took a different approach to the chances that Julie wins an individual game. Given that badminton is a very skillful game, I think the long-run win percentages for Julie vs. John on a given game could range from 1% to 99%, and still satisfy the constraint that both are good players. I defend this assumption by the fact that the 100th best tennis player in the world is a "good player." Yet, he's probably about 1% to beat Djokovic in a match.

So, I started with Julie having a 1% chance to win a game, and then simulated 100,000 seven game sets and 100,000 nine game sets. Then upped her percentage to 2%, etc. all the way up to 99%. So I simulated 9.9M games for each of 9 game sets and 7 game sets, over a wide range of win percentages.

I get the same answer as to which is more likely. But very different overall percentages:

 

Chances of winning exactly 4 out of 7 games: 12.6%

Chances of winning exactly 5 out of 9 games: 10.1%

BTW, if Julie is exactly as good as John, I agree that she will win exactly 4 out of 7 game about 27% of the time.