Best-of-7 Series: 3-0 Comebacks

Since it's NBA playoffs season, a common fact that gets thrown around is that no NBA team has ever come back from a 3-0 deficit to win a best-of-7 series. Of course, the NHL and MLB also employ a best-of-7 format (though the MLB only does it for 3 series a year), and both of these leagues have had (at least) one team come back from a 3-0 hole. 

Through fall 2014, NHL teams have performed such a comeback 4 times out of 178 tries (2.25%), NBA teams have come back 0 times out of 110 (0%), and MLB teams (go Red Sox!) have come back 1 time out of 34 (2.94%), for an overall rate of 5 times out of 322 (1.55%). [1]

I thought I'd go through a quick exercise on how likely it is for teams to come back from a 3-0 hole in a best-of-seven series. 

1. Warmup: Team A has a $50$ percent chance of winning any particular game in a best-of-7 series. What is the probability that Team A loses the series, given that it wins the first three games?
2. Team A has a $p$ chance of winning any given game in a best-of-7 series. What is the probability that Team A wins the first three games?
3. Now we don't know anything about $p$, other than the fact that it is drawn from a uniform distribution on $[0,1]$. What is the probability that Team A wins the first three games?
4. Again, $p$ is drawn from a uniform distribution on $[0,1]$. What is the probability that Team A wins the first three games but then loses the next four games?
5. Given that $p$ is drawn from a uniform distribution, what is the conditional probability of Team A losing the series given that it wins the first three games?

If you work these exercises out correctly, you should have gotten the following answers:

1. $\frac{1}{16}$
2. $p^3$
3. $\frac{2}{5}$
4. $\frac{1}{315}$
5. $\frac{1}{126}$

Under this model, the probability of NBA teams going 0-for-110 is $\left(\frac{125}{126}\right)^{110}$, or about 41.6%, so (given this model) it's not too surprising that such a comeback has never happened. 

In reality, NBA/NHL/MLB teams in the playoffs are usually more evenly matched than this model would suggest: for example,  you don't ever see an MLB team being given less than a $\frac{1}{3}$ chance of winning a particular game. Therefore we would expect the distribution for $p$ to be much more concentrated around $\frac{1}{2}$ (compared to the uniform distribution). This in turn should increase the conditional probability of coming back from a 3-0 deficit. The extreme case of this would be when $p$ is fixed at $\frac{1}{2}$, and as we saw there, the condition probability of a 3-0 comeback gets as high as $\frac{1}{16}$.

This model also doesn't account for the home/away split, but my intuition is that that would probably also increase the conditional probability of such a comeback happening. Indeed, the overall comeback rate for 3-0 series is higher than what this simplified model predicts.

In any case, we should be probably be surprised that a 3-0 comeback has never happened in the NBA, but at the same time, don't expect one to happen any time soon.

[1] http://www.whowins.com/tables/up30.html