This morning, after I made some DaDakota-esque predictions about how the Rockets could (and would) win the series, my buddy e-mailed: "Elisha Cuthbert could be waiting for me when I get home, but it's about as likely as the Rockets winning three in a row to win the series." There's a logical, statistical fallacy here. And my friend was just joking around, but I see the media make the same mistake. The fallacy is to apply the slim "down 3-1 odds" to the current odds of us winning the series. It's like a coin flip. Let's say, hypothetically, you flip a coin nine times in a row, and you get heads all nine times. Crazy, right? But possible. For that tenth flip, it's a fallacy to say, "The odds of the coin being heads 10 times in a row are really low, so it's probably gonna be tails." The odds of that tenth flip are still fifty/fifty. I get that it's not a perfect comparison, as, unlike coin flips, basketball games are not independent, discreet actions; a win or loss can influence the odds of a subesquent win or loss. But still, the overall point stands. The "3-1 scenario" is now outdated, old data. Just like the first 9 coin-flips are irrelevant in that tenth flip's probability. The Rockets don't have to worry about winning three games in a row. They just need to win two. And if they win the first, they'd be favored in the second. So the next time some pundit tells you "Teams down 3-1 have rarely, rarely won the series," think about that tenth coin flip. I realize this isn't really new information, but somehow this cold, mathematical reasoning makes me feel better about our chances, about the series.
Statistically speaking, you are 100% correct. So I hope even more that the Rockets win Friday! *GO ROCKETS*
Yes, but which bet do you want to take? 1) You bet me $10 that the next flip of the coin is heads. 2) You bet me $100 that the next 10 flips of the coins are heads.
Well, our starting PG was out the first 2 games. That really has made a significant difference. Since he's been back, we've won 2 out of 3. So, the team playing this next game is 2-1, not 2-3.
I'll take #1, please. The odds of #1 are 50/50, so it's a reasonable bet. The odds of #2 are 0.5 to the tenth power, and yes, I'm a geek, that works out to be 0.098%. Not a good bet. (Much more likely, though, than our 22-game winning streak.) Anyway, I'm not arguing that mathematically it's probable that we'll win the series. If you just apply a basic 50% chance for games 6 and 7 (which of course isn't the case), technically, the odds are only 25% that we win both games. (50% x 50%). The point is that without realizing it, people are intuitively "punishing" our chances by thinking about that earlier 3-1 deficit. The ten coin flips is just an exaggerated way to illustrate the fallacy -- like after the 9 heads it's tempting to think "man, no way it can be 10 in a row - we're due for a tails." Nope. The fact that you flipped 9 consecutive heads is now irrelevant.
Your assumption holds only IFF each team's expected winning probabilities are equal. However, in reality, we do NOT know the true winning probabilities of each team, so we have to find sample winning probabilities and infer the true winning probabilities from the samples. We know that the difference btw the sample expected win% and true expecte win% => 0 whenever n=> infinity. We have a limited number of games n=7, so we normalize to that assumption. Then a 1-3 record => 25% winning with n=4 is closer to the true winning% than a 1-1 record => 50% winning with n=2games. The sports analysts' intuitive reasoning that a team with a 1-3 record has a very low probability of winning also includes this knowledge that a team with a 1-3 record has higher likelihood of having a winning probability < 50%. This explains why there have been so few teams that can pull out a win after being down 1-3. It usually implies that they have a lower winning % w.r.t to the other team i.e. team 1 is inferior to team 2.
Just tell me the chances of us winning 22 games in a row including 10 of those without Yao. Impossible is nothing.
It's not even about to win 2 games in a row. It's about winning the next game first. That's it. Just one game. We can do it.
Rockets vs. Jazz is not a coin flip though. Comparing our 22 game streak to 22 coin flips doesn't work either. What if the Jazz are better than the Rockets and every game they have, say, a 63% chance of winning? Albeit improbable that the streak went on that long, it is not like the odds were 50%^22. We were better than most of those teams. You are dead on when you say that the statistics reset after each game.
Aren't you assuming that each game is independent of eachother? In the playoffs, teams make adjustments. So, when you're facing a tough, well-coached opponent, it is difficult to string together consecutive wins.
It's the theory of relative probability. If you have 3 doors to choose from with one having 1000 dollars behind it, A B and C.. and you choose A.. Now someone tells you that C is not the door, would you change your pick? Your answer should be yes. B/c at the beginning if you were given the choice of A or B and C, knowing C isn't the door, you have 67% chance of winning as opposed to 33%. Therefore, b/c of past odds, it is better to change your pick to B, gaining you another 34%. Now in basketball, b/c of its in-objectivity, the coin flip fallacy doesn't really apply. Sure it's 50/50 for each game, but home court, momentum, etc. all make a difference.
In addition, they also say that b/c teams down 3-1 have 3 games to lose, where as the other team needs 1 game to win. What are the odds of a coin landing heads 3 times in a row compared to tails 1 time? Catch the drift?
Its called the Gambler's fallacy, and you are correct for the most part, however, with basketball where players can decide the outcome, unlike a coint tos, the same 50/50 probability does not necessarily apply, hence the reason for sports analysts.
I think I did a lousy job explaining my thinking. Of course the odds aren't 50/50. Of course the odds of winning one game can be affected by the prior game. But.... the core thesis is that when (negatively) assessing the Rockets chances to win the series as of today, they're intuitively using the low probability of their chances to win as of being down 3-1. This is the Gambler's Fallacy (thanks, tonyrox, that's the right label I was forgetting.) My point is that we have a better chance to win than most give us credit for, because they're letting the 3-1 deficit color their perception. (Just as the 9-in-a-row tails coin flip would make someone think that the tenth should be tails)
Yeah, this isn't probability. We are heavy underdogs in the first game, and even if we win that, Vegas may still play the second game as a coin flip. Game 6 factors: 1. Utah's extreme home court advantage 2. Their 3-2 lead in the series 3. The Rockets having won two out of the five times anyone has won in Utah this year 4. The Rockets having won the last game by a wide margin and playing relatively pressure-free. Conclusion: Utah is heavily favored. One site has them at -7.5 (projected to win by 7.5), with 2:1 odds for anyone betting on the Rockets. Potential Game 7 factors: 1. Game being in Houston, with a moderate home court advantage. 2. The 3-3 tie in the series. 3. The Rockets having momentum with two wins in a row. 4. The mental composure effect: Utah won at Houston in Game 7 last year, McGrady has never won a playoff series yet, and Houston has not won a series since '97. Vegas will not forget that last factor. If there is a Game 7 it could be as close to a coin toss as you can get.
