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Old 09-08-2017, 11:41 AM   #1
DMMx69
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Odds of a 15 game win streak given 50 games to do it?

http://www.espn.com/mlb/story/_/id/2...eak-reaches-15

Guy pays $75k to insure against a loss of $1.7MM

The insurance company pays out the 1.7MM if the Cleveland Indians won 15 games in a row, any time after August 1st.

So, something like, they had 50 games left, and needed a 15-game win streak.

If they were 60% to win each game (I know not accurate, but for sake of simplifying problem), what is the chance they have a 15-game win streak?

.6^15 = .00047, the chance of them winning 15 in a row at any specific point. They had 36 opportunities to start such a streak, but I'm not sure if multiplying that by 36 is right, since the opportunities overlap?

Any help would be great. Thanks in advance.
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Old 09-08-2017, 03:13 PM   #2
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Re: Odds of a 15 game win streak given 50 games to do it?

Win Streak Calculator

https://www.google.com/search?q=win+...e=utf-8http://
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Old 09-08-2017, 03:32 PM   #3
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Re: Odds of a 15 game win streak given 50 games to do it?

We typically want to know the probability that a streak of at least a certain length occurs in a given horizon. For example, if the Indians win 16 games in a row, that is also considered a 15-game winning streak.

Due to the overlap issue that OP rightly points out, the derivation of the formula for this probability is a bit complex. The formula is available on the internet, of course. It essentially boils down into a polynomial in terms dealing with the win prob raised to the Kth power (where we seek K wins in a row).

If I plugged in the correct terms into the formula, I get that the probability of winning at least 15 games in a row out of a total of 50 games, where the win prob of each game is independently given as 60%:

= (15)*(.6)^15 - (38.4)*(.6)^30 + (2.24)*(.6)^45

= .0070443
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Old 09-08-2017, 03:47 PM   #4
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Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
Originally Posted by whosnext View Post
If I plugged in the correct terms into the formula, I get that the probability of winning at least 15 games in a row out of a total of 50 games, where the win prob of each game is independently given as 60%:

= (15)*(.6)^15 - (38.4)*(.6)^30 + (2.24)*(.6)^45

= .0070443
That's what a streak calculator shows.
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Old 09-09-2017, 07:28 AM   #5
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Re: Odds of a 15 game win streak given 50 games to do it?

Wonder if the person who wrote the policy gets a raise for the insane markup or fired due to the result.
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Old 09-09-2017, 02:27 PM   #6
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Re: Odds of a 15 game win streak given 50 games to do it?

I admit I have gone down the rabbit hole exploring this interesting problem. It is remarkably both simple and complex. There are multiple approaches to solving the problem, including Markov Chains.

Anyway, I have fiddled around with one of the solution formulas I found on the internet. I have rearranged terms and obtained a fairly straightforward result:

Let P = the prob of winning each single game.
Let Q = 1 - P
Let N be the total number of games in which the streak can occur.
Let K be the number of games in the streak we are interested in.
Let W = P^K
Let Z = [N/K], the integer value of N/K.
Let C(X,Y) be the number of ways Y items can be chosen out of X total items (the choose function).

Then the prob of observing a winning streak of K or more games in N total games:

= W[1+(N-K)*Q] + Sum [T goes from 2 to Z] (W^T)*((-1)^(T-1))*[C(N-T*K,T-1)*(Q^(T-1)) + C(N-T*K,T)*(Q^T)]
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Old 09-14-2017, 11:10 PM   #7
IOWActuary
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Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
Originally Posted by whosnext View Post
We typically want to know the probability that a streak of at least a certain length occurs in a given horizon. For example, if the Indians win 16 games in a row, that is also considered a 15-game winning streak.

Due to the overlap issue that OP rightly points out, the derivation of the formula for this probability is a bit complex. The formula is available on the internet, of course. It essentially boils down into a polynomial in terms dealing with the win prob raised to the Kth power (where we seek K wins in a row).

If I plugged in the correct terms into the formula, I get that the probability of winning at least 15 games in a row out of a total of 50 games, where the win prob of each game is independently given as 60%:

= (15)*(.6)^15 - (38.4)*(.6)^30 + (2.24)*(.6)^45

= .0070443
What do the powers of 30 and 45 mean in laymens terms? Why does it work out that way?
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Old 09-14-2017, 11:51 PM   #8
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Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
Originally Posted by IOWActuary View Post
What do the powers of 30 and 45 mean in laymens terms? Why does it work out that way?
I don't have a good answer, so I will link to the derivation of the exact formula.

http://www.askamathematician.com/201...swer-is-known/

If anybody wants to chime in here, go for it.
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