Question

Your Toronto Maple Leafs won 30 of 82 games last season (i.e., the 2014-2015 season), giving them a winning percentage of 37%. If we assume this means the probability of the Leafs winning any given game is 0.37, then we can predict how they would have done in a playoff series.

Answer the following questions to determine the probability that the Leafs would have won a best of 7playoff series (i.e., won 4 games) had they made the playoffs last season.

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).)

c. Calculate the number of sequences as they relate to this problem. (Note that the answer is not 27 as not all series would last 7 games.)

d. Calculate the probability that the Leafs would win the series.

e. What is your best guess for the probability that the Leafs will ever win the Stanley Cup again (the ultimate prize in the NHL)

Answer 1

a)

If we let 1 represent a win, and 0 represent a loss, the fastest win is 1 1 1 1, and theres only one way to achieve this. The longest win is 1 0 1 0 1 0 1, and there is C(7,4)=35 ways to achieve this. Thus the longest is 35 sequences, and shortest is 1 sequence.

b)

In order to win, they must win 4 games. Thus we have to consider when the games only go to 4, to 5, to 6, and to 7. That is 1 + C(5,4) + C(6,4) + C(7,4) = 56

(c)

we can count the number of ways the Toronto Maple Leafs can win.

Toronto Maple Leafs wins in 4: 1

Toronto Maple Leafs wins in 5: (43)(43)

Toronto Maple Leafs wins in 6: (53)(53)

Toronto Maple Leafs wins in 7: (63)(63)

We add these 4 results together and multiply by 2, accounting for the number of ways the other team can win.

For (d) you would find the probability of each happening:

P(Toronto Maple Leafs wins in 4): .374.374

P(Toronto Maple Leafs wins in 5): (43)(43)(.373.373)(.63)

P(Toronto Maple Leafs wins in 6): (53)(53)(.373.373)(.632.632)

P(Toronto Maple Leafs wins in 7): (63)(63)(.373.373)(.633.633)

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