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 09-08-2017, 11:41 AM #1 DMMx69 old hand     Join Date: Jul 2006 Location: This Mortal Coil Posts: 1,678 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.
 09-08-2017, 03:13 PM #2 statmanhal Pooh-Bah   Join Date: Jan 2009 Posts: 3,930 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://
 09-08-2017, 03:32 PM #3 whosnext Pooh-Bah     Join Date: Mar 2009 Location: California Posts: 3,778 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
09-08-2017, 03:47 PM   #4
statmanhal
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Join Date: Jan 2009
Posts: 3,930
Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
 Originally Posted by whosnext 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.

 09-09-2017, 07:28 AM #5 Monteroy Carpal \'Tunnel   Join Date: Dec 2006 Posts: 8,947 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.
 09-09-2017, 02:27 PM #6 whosnext Pooh-Bah     Join Date: Mar 2009 Location: California Posts: 3,778 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)]
09-14-2017, 11:10 PM   #7
IOWActuary
enthusiast

Join Date: Jul 2014
Posts: 97
Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
 Originally Posted by whosnext 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?

09-14-2017, 11:51 PM   #8
whosnext
Pooh-Bah

Join Date: Mar 2009
Location: California
Posts: 3,778
Re: Odds of a 15 game win streak given 50 games to do it?

Quote:
 Originally Posted by IOWActuary 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.

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

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