Probability Help

[henrix77]

I must first preface the question with an apology if this is incredibly standard...I'm not a math expert by any means, so I figured if I wanted a good answer, I'd come to those who were.

Anyway, suppose you are playing a game in which you have a 45.5% chance of a win, a 45.5% chance of a loss, and a 9% chance of a tie. For the sake of argument, let's assume that everyone playing this game has these odds as everyone plays optimally.

Now, what are the odds that over a sample of 150 games, you'd see two six game losing streaks, and also an eleven game losing streak? I'm not only curious about the answer, but the method used to find it. Any help is appreciated. Thanks.

[henrix77]

Just a bump...

No one knows how to do this? Bummer. Still appreciate any input anyone can offer.

[ncray]

I've been drinking a little, so I'm just going to do the eleven game losing streak. I'm also going to assume that this is 11 consecutive losses (not, say loss + tie + ten more losses).

The probability of losing 11/11 at 45.5% is just .455^11, or 0.000173032. Now, to find the probability that you get an 11 loss streak in 150 games, start by considering the case when you start off losing 11 straight games. You'd have to win or tie the 12th, otherwise it would be a 12 game losing streak.
Case 1: lllllllllll(w or t)...
Probability: .455^11*.545*1 = 0.0000943022
Case 2: (w or t)lllllllllllllll(w or t)...
Probability: .545*.455^11*.545*1 = 0.0000513947
Cases 2 through 140 have the same probability, since you flank the 11 game losing streak with ties or wins on either side, and case 1 has the same probability as case 140 which is ...(w or t)lllllllllll

So, our probability of having at least one eleven game losing streak in a game is 2*0.0000943022 + 139*0.0000513947 = 0.00733247, or .73%. Note that the answer to the question "What is the probability of having at least one eleven game or more losing streak?" would be higher.

[henrix77]

Thanks!

I appreciate the input. If anyone else has anything to offer, specifically, showing further what the odds of having TWO six game losing streaks ALONG WITH that eleven game losing streak within those 150 games, that would be awesome. Thanks again.

[T50_Omaha8]

[ QUOTE ]
Thanks!

I appreciate the input. If anyone else has anything to offer, specifically, showing further what the odds of having TWO six game losing streaks ALONG WITH that eleven game losing streak within those 150 games, that would be awesome. Thanks again.

[/ QUOTE ]The problem with this question is that any fairly specific scenario will have a very small probability. I could ask "what is the probability of a 5 game losing streak, a 7 game losing streak, and an 11 losing streak" and it identifies a completely different event, also with a tiny probability.

It's like asking the odds getting dealt 72 offsuit and the flop coming 357 all of spades and the turn and river coming aces--it's just another small, unlikely event. It's better to ask open ended questions like "what is the chance of having a losing streak of 11 or more games?" as they yeild more useful results.

Sorry to be a bummer.