Negative Binomial Distribution for Variable Probabilities


I wonder if someone can help me.

I'm looking to estimate the probability of player a v player b winning a darts match where the player who wins 6 legs first wins the match. I would normally use a negative binomial distribution for this kind of situation. However, darts is complicated because the player who throws first alternates between the players after each leg and this changes the probability of the leg outcome.

For example:

Leg 1: Player A throws first (Player A 63% win chance, Player B 37% win)
Leg 2: Player B throws first (Player A 45% win chance, Player B 55% chance)
Leg 3: Player A throws first (Player A 63% win chance, Player B 37% win)
Leg 4: Player B throws first (Player A 45% win chance, Player B 55% chance)

And so on. Any help would be hugely appreciated as I'm banging my head against a brick wall!


Ambassador to the humans
Negative binomial wouldn't work. I don't know of a 'nice' way to do this. There are a few options though but all require some sort of enumeration of what could possibly happen.

My initial thoughts were: 1) just to enumerate all the possibilities 2) calculate the probability of each possibility and then 3) sum the probabilities of each possibility that meets the criteria. This is easy enough to do using a programming language.

I then thought that one could use the fact that "first to six" is the same as "best of 11". So if we can just figure out the probabilities of A winning 0, 1, 2,... 11 in a set of 11 games then sum the probabilities for >= 6 we will get our answer. We can use excel and some relatively simple formula manipulation to do this relatively quickly by recognizing that in round X the probability that A has won Y games is: P(A won Y-1 games in round X-1)*P(A wins this round) + P(A win Y games in round X-1)*P(A loses this round).

I made a spreadsheet in excel to do just that. You can control the probabilities that "A" wins each round so you could even play around with having an entirely different probability for each round instead of just flip-flopping probabilities between rounds. For your example the answer that A wins comes out to be 0.630551086 but I suspect you don't care about that particular example and are more interested in how to get the answer so you could play around with what happens for different probabilities.

Thanks so much for this answer. Apologies if I'm misunderstanding but does this account for the fact that in reality, once a player reaches 6, no further legs are played? The only reason I ask is that the spreadsheet goes up to 11 games for player A.

Probably a stupid question. Apologies if so!



Ambassador to the humans
It does and it doesn't. It doesn't stop play but that doesn't matter. Once somebody hits six wins they can't go on to lose. So basically what it does is say that stopping at six wins and playing the full eleven games is the same thing in terms of figuring out who wins the series. We could program in rules so that it stops once somebody gets six wins but it would give the same answer but take more work.

And it's not a stupid question. If it's not immediately obvious why it ends up being this way I suggest working out all possibilities in a slightly easier example - first to two wins (so best out of three). There are only eight possible ways that can play out so it's easier to enumerate and play with. But you should be able to convince yourself that it gives the same results either way.
I've had a play around and I am convinced that it works. I kind of get why although have to admit I did find it a little confusing.

A big thanks, hugely appreciated. Not sure I would have had a clue how to do this without mapping out all the possible combinations so you're saved me big time!

Thanks again! :)
Sorry another question has come to mind. Is it possible to use this sheet to identify the probabilities of each combination of leg wind.

E.g. Probability of player a winning 6-0, 6-1 etc?

Thanks in advance.


Ambassador to the humans
Indirectly sure. You could use the round six column and look at six wins for A to get that. To get 6-1 victory you couldn't just look at round 7 with six wins for A because that would include the probability that A gets six wins in the first six wins and loses round seven. But to win 6-1 they HAVE to be 5-1 in round six. Do take that value and then multiply by the probability of victory in round seven.

Similarly to get probably of 6-2 victory you could look at probability of 5-2 in round seven and multiply by probability of win in round 8. Hopefully the pattern is clear.