[Fruriourl]

I saw this news item about the Bryan Brothers Davis Cup win and it got me thinking:

"The USA has only once recovered from a 0-2 deficit -- in 1934 -- but its record improves when down 1-2, having done it five times with the last being in 2000."
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I was wondering, have our chances really improved? At first glance I would think the doubles match is irrelevant, in order to come back from 0-2, both singles players have to flip to 2-0. Whatever happens in the double match has no impact on how the singles players perform. Obviously if the Bryan's lost we'd be out, but the guys who went 0-2 don't play doubles, so theoretically those are two independent events.

Maybe a stats/information theory guy can help me out on this.

[SkatrySkith]

jaundixonformpv,
"Have our chances really improved?" No, not really. While winning two out of two matches is easier than winning three out of three, in this instance it won't matter.
The main way of winning when down 1-2 is having won one of the singles matches. Since the top player from each team plays the second best player on the other team, the score after the first day is often 1-1. So that Team A's Player 1 beats Team B's Player 2, and Team B's Player 1 beats Team A's Player 2.
Then if Team A's doubles team wins, Team B can hope that the actual ordering of players strengths is Team B Player 1 > Team A Player 1 > Team B Player 2 > Team A player 2. In this case, Team B can be expected to win both Sunday singles matches. In this case, however, the ordering is Spain Player 1 > Spain Player 2 > USA Player 1> USA Player 2.
If the Bryan brothers had lost, we (USA) would have lost aready, and our chances of winning would be zero. Since the Bryan brothers did win our chances of winning have increase enough to actually match the odds of the poverbial snowball.

[noingenah]

Maybe I don't understand the question... but it seems to me winning 3 of 3 is harder than winning 2 of 2. Are the odds better if the US now wins the first singles match tomorrow? Of course.

[JulieSmithdccd]

It took a couple reads but I do understand it now. The doubles match would likely be the most independent of all five matches, so this seems to make sense to me. If you have already won a singles match (ie you were 1-1 in singles) that would mean you're more likely to go 2-0 on the third day than if you had gone 0-2 the first day - you obviously have at least one person capable of winning singles matches.

Wonder how this stat works out for other countries?

[dodsCooggipsehome]

Mathematically (I think):
You can split the tie in three groups
Day 1 (two singles)
Day 2: (one doubles)
Day 3: (two single).

Assuming every match is 50/50, then for day one your odds are:
25% you will be up 2-0
50% you will be 1-1
25% you will be down 0-2

For day 2, you only have 50% of going 1-0, 50% going 0-1
Day three is the same as day 1, of course.

So, if by the end of day 1 you are down 0-2, your chances (again at random) of getting the next three matches are 50% (1-0 in doubles) times 25% (2-0 in the last two singles).
0.5*.025 = .0125

12.5% (or 1/8) chances of coming back from 0-2, if matches are at random.

If by the end of day 1, you are 1-1, then your chances of getting 2 of the next three are 2/3, 66% (if matches are at random)

It is as somebody said. A whole different scenario to be 1-2 because you split the first day matches, than 1-2 because you won the doubles.

[jagxj12]

If by the end of day 1, you are 1-1, then your chances of getting 2 of the next three are 2/3, 66% (if matches are at random)

That is clearly not so - both teams have 50% chance of winning.

Mathematical formula for determining probability of event happening is given by this formula:

Pr = (n/k)pkqn-k where n = times played (3 in our case), k = wins needed (2), p = probability of success (random so 0.5), q = probability of failure (also 0.5).

This will give 0.375, however that is a probability of winning exactly 2 matches. Since 3 out of 3 is also good for our cause you'll have to add the probability of that (0.125) giving the final and rather obvious answer of 50% chance.

You can also use this webpage (binomial trials calculator) to do all kinds of more complex probability calculations.