SOLUTION: 1. The Knick play the Bulls in a seven game series that end when one team has won four games. We record the outcome of a game with a W for a Knick win and an L for a Knick loss.

Algebra ->  Permutations -> SOLUTION: 1. The Knick play the Bulls in a seven game series that end when one team has won four games. We record the outcome of a game with a W for a Knick win and an L for a Knick loss.       Log On


   

Question 917652: 1. The Knick play the Bulls in a seven game series that end when one team has won four games. We record the outcome of a game with a W for a Knick win and an L for a Knick loss.
How many series would have to be played to be sure that the same outcome happens twice?
I know there are a total of 840 possible outcomes from a permutation of (7,4), so would it just be 840 divided by 2?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You have seven games. There are 2 outcomes for each game.

There are 2^7 = 128 ways to have a Ws and Ls in a seven character sequence; however, you cannot have something like WWWWWLL because once you get that 4th W, the series is over. So in reality, you the upper limit on Ws is 4. The same is said for the L's as well.

So here is a list of all the possible outcomes.

For example, scenario #3 has "LLLWWL" under the "Game Sequence" column. This means we have the Knicks lose 3 times in a row (L, L, L). The next game has the Knicks winning (W). The Knicks win again in game 5 (W). However, the Knicks lose (L) game 6. Overall, they lose the championship series because the opponent has gained 4 wins before them. This is what I mean by "Knicks lose" in the "Championship Summary" column.

The other rows are read in a similar way. Let me know if you have any questions on that.

There are 70 possible outcomes.


Number Game Sequence Championship Summary
1 LLLL 4 game series, Knicks lose
2 LLLWL 5 game series, Knicks lose
3 LLLWWL 6 game series, Knicks lose
4 LLLWWWL 7 game series, Knicks lose
5 LLLWWWW 7 game series, Knicks win
6 LLWLL 5 game series, Knicks lose
7 LLWLWL 6 game series, Knicks lose
8 LLWLWWL 7 game series, Knicks lose
9 LLWLWWW 7 game series, Knicks win
10 LLWWLL 6 game series, Knicks lose
11 LLWWLWL 7 game series, Knicks lose
12 LLWWLWW 7 game series, Knicks win
13 LLWWWLL 7 game series, Knicks lose
14 LLWWWLW 7 game series, Knicks win
15 LLWWWW 6 game series, Knicks win
16 LWLLL 5 game series, Knicks lose
17 LWLLWL 6 game series, Knicks lose
18 LWLLWWL 7 game series, Knicks lose
19 LWLLWWW 7 game series, Knicks win
20 LWLWLL 6 game series, Knicks lose
21 LWLWLWL 7 game series, Knicks lose
22 LWLWLWW 7 game series, Knicks win
23 LWLWWLL 7 game series, Knicks lose
24 LWLWWLW 7 game series, Knicks win
25 LWLWWW 6 game series, Knicks win
26 LWWLLL 6 game series, Knicks lose
27 LWWLLWL 7 game series, Knicks lose
28 LWWLLWW 7 game series, Knicks win
29 LWWLWLL 7 game series, Knicks lose
30 LWWLWLW 7 game series, Knicks win
31 LWWLWW 6 game series, Knicks win
32 LWWWLLL 7 game series, Knicks lose
33 LWWWLLW 7 game series, Knicks win
34 LWWWLW 6 game series, Knicks win
35 LWWWW 5 game series, Knicks win
36 WLLLL 5 game series, Knicks lose
37 WLLLWL 6 game series, Knicks lose
38 WLLLWWL 7 game series, Knicks lose
39 WLLLWWW 7 game series, Knicks win
40 WLLWLL 6 game series, Knicks lose
41 WLLWLWL 7 game series, Knicks lose
42 WLLWLWW 7 game series, Knicks win
43 WLLWWLL 7 game series, Knicks lose
44 WLLWWLW 7 game series, Knicks win
45 WLLWWW 6 game series, Knicks win
46 WLWLLL 6 game series, Knicks lose
47 WLWLLWL 7 game series, Knicks lose
48 WLWLLWW 7 game series, Knicks win
49 WLWLWLL 7 game series, Knicks lose
50 WLWLWLW 7 game series, Knicks win
51 WLWLWW 6 game series, Knicks win
52 WLWWLLL 7 game series, Knicks lose
53 WLWWLLW 7 game series, Knicks win
54 WLWWLW 6 game series, Knicks win
55 WLWWW 5 game series, Knicks win
56 WWLLLL 6 game series, Knicks lose
57 WWLLLWL 7 game series, Knicks lose
58 WWLLLWW 7 game series, Knicks win
59 WWLLWLL 7 game series, Knicks lose
60 WWLLWLW 7 game series, Knicks win
61 WWLLWW 6 game series, Knicks win
62 WWLWLLL 7 game series, Knicks lose
63 WWLWLLW 7 game series, Knicks win
64 WWLWLW 6 game series, Knicks win
65 WWLWW 5 game series, Knicks win
66 WWWLLLL 7 game series, Knicks lose
67 WWWLLLW 7 game series, Knicks win
68 WWWLLW 6 game series, Knicks win
69 WWWLW 5 game series, Knicks win
70 WWWW 4 game series, Knicks win



Again there are 70 possible outcomes. The other 128-70 = 58 outcomes are not possible. This would be something like WWWWWLL (5 W's aren't possible).


I didn't use a formula. I simply listed out all of the possible outcomes (all 128) using this combinatorics calculator. Then I picked out the outcomes that weren't possible.

So this means that 71 series would have to be played to make sure the same outcome happens twice. They need to play 70 series to generate all of the outcomes. That 71st game will be a repeat of one of the 70 already listed.