What is the Probability of the Toronto Maple Leafs Winning a Playoff Series?

[Mouse07]

Homework Statement

Your Toronto Maple Leafs won 45 of 82 games last season giving them a winning percentage of 55%.
If we assume that this means the probability of the Leafs winning any given game is 0:55, then we
can predict how they may have done in a playoff series. Answer the following questions to determine
the probability that the Leafs would have won a best of 7 playoff series (i.e., won 4 games) had they
made the playoffs.

(a) Rephrase this question in terms of sequences of 0s and 1s. What is the shortest length of a
sequence? What is the longest length of a sequence?

(b) Calculate the number of sequences which correspond to the Leafs winning the series. Note that
the answer is not C(7; 4).

Homework Equations

The Attempt at a Solution

need help on b i get 35 but i don't use c(7;4) directly, i am unsure if i am using it in this calculation,
4-1 series is 4C1. In a 4-2 series 5C2 = 10. Finally, for the 4-3 series 6C3 = 20.
and up for total number of sequense they can win the series

 

[mfb]

Don't forget 4-0. The sum of all should give the correct answer.

 

[Mouse07]

ya i forgot to write i did that too, but when i add i get 35 which is same as 7c4 so i am confused if i used it or not.

 

[Ray Vickson]

ya i forgot to write i did that too, but when i add i get 35 which is same as 7c4 so i am confused if i used it or not.

Please try to write in legible and proper sentences, using capital letters, etc.

Anyway, the answer is the same (that is, C(7,4)) because the number of ways in which TO can win the series is the same, whether they play all 7 games or whether they stop upon winning 4 games. In other words, they could win the first 4, then play 3 more useless games, without changing the fact that they win the series.

WARNING: the *probabilities* are not the same here! To get the probability that TO wins you cannot just look at the probability they would win 4 out of 7; you need to use the fact that the play stops whenever either team reaches 4 wins (and so, for example, the other "useless" games would NOT be played). Maybe that is what the question meant when it said the number is not C(7,4)

 

[mfb]

C(7,4) written as string of 1 and 0 contains strings like 1111000 which end with losses - you can interpret them as 1111, so you get the same number. But following this logic, you could add things like 1111111, too, and then you get problems with the number of different series. In addition, as Ray Vickson mentioned, the probability for 1111 is not the same as the probability for 1111000, and you should keep those things separate.

 

[Ray Vickson]

I should have said that after winning the first 4 the team can play and *lose* the next 3 games, etc. That is, they still win exactly 4 games.

 

[Mouse07]

oh okay, and i will try to write proper sentences.

and one more thing can i count the probabilities using Bernoulli trial,
for each scenario like... what is probability of Team win the series

team wins series 4-0 P= p^4
team wins series 4-1 P = 4 p^4 q
team wins series 4-2 P = 10 p^4 q^2
team wins series 4-3 P = 20 p^4 q^3

and then add all of the probabilities.

Thank you for helping :)

 

[Ray Vickson]

oh okay, and i will try to write proper sentences.

and one more thing can i count the probabilities using Bernoulli trial,
for each scenario like... what is probability of Team win the series

team wins series 4-0 P= p^4
team wins series 4-1 P = 4 p^4 q
team wins series 4-2 P = 10 p^4 q^2
team wins series 4-3 P = 20 p^4 q^3

and then add all of the probabilities.

Thank you for helping :)

win on 4-0: prob. = p^4
win on game 5: must win 3 of the first 4, then win game 5, so prob. = C(4,3) p^3 q p = 4 p^4 q, so yes, your formula is correct in this case. The others are the same, but you should write out the reasons, not just the final formulas---after all, you want whoever is marking your work to understand what you are doing and give you the marks that you deserve.

You said you were going to use proper sentences, then just went ahead and ignored your own promise.

 

[Mouse07]

To be honest, i am not that great in explaining things online.
I really appreciate your help and i hope i can improve and keep my promise.

thank you :)