The Patriots and the Conjunction Fallacy

The conjunction fallacy is when people judge the probability of a series of events to be larger than one probability of its component events. In simple terms, many people mistakenly assign a higher probability to a specific outcome than a more general one. In football terms, this means that many fans underestimate how difficult it is for a team, even an extremely good one, to win the Super Bowl.

In my very simple poll asking if the Patriots had a better than 50/50 chance to win the Super Bowl, 64% of the (few) respondents said yes. I'd vote no, but let's look at what it would take for NE to win the championship.

The Patriots need to win three consecutive games, two of which are at home and one at a neutral site, against the NFL's top teams. What kind of win probability would they need for each game to arrive at a 50/50 chance to win it all? x * y * z = 0.50. For a rough estimate, let's assume their chance to win each game is roughly equal. Their probability would need to be 0.79 in each game. (0.79^3 = 0.50.)

This seems reasonable, but since they wouldn't have home field advantage in the Super Bowl, they would need slightly higher probabilities for the division and conference rounds of the playoffs.

Let's look at what Vegas thinks. According to a major online gambling site, NE is given 9 to 4 odds (0.69 probability) of winning the AFC championship, and 3 to 2 odds (0.60 probability) of winning the Super Bowl. They are also 13 point favorites to beat JAX this Saturday. With a 49 point over/under, 13 points roughly equates to a 0.78 probability (using this method).

There is something out of whack. A 0.60 probability of winning the Super Bowl and a 0.69 probability of winning the AFC, means the individual Super Bowl game probability must be 0.60/0.69 = 0.87. That's amazingly high. And I suspect that's where the conjunction fallacy may be having an effect. The individual game odds are incongruent with the conjunctive odds of NE winning all three games.

I'd guess there is some kind of arbitrage opportunity there for gamblers. Personally, I just think it's interesting how some people intuitively estimate the odds of future events.

My own model estimates NE has a 0.74 probability of winning this weekend. Against IND they get a 0.65 probability. But against SD they would have a 0.84 probability of winning. In total, that gives NE a 0.68 chance at appearing in the Super Bowl. To have a 50/50 chance of winning the Super Bowl, NE would need a 0.74 chance of beating the NFC representative. That would be the same chance they have against the AFC's #5 seed, a (pretty good) Florida team playing in Foxboro in January.

My own sense is that the Patriots have about a 40-45% chance of winning the Super Bowl. I would say 40, but as a dome team, the Colts would have a tougher time in Foxboro due to the January weather.

4 Responses to “The Patriots and the Conjunction Fallacy”

1. j holz says:

Maybe I'm not reading it right, but doesn't Phil Birnbaum decry that exact method of calculating win% in the post that you linked to?

I think this weekend's moneyline is a much better indicator of NE's chances of winning, and they're 7-1 favorites there. But I do agree that the futures lines offer an arbitrage opportunity.

2. jaginma says:

I may be off base, but when the odds makers set a money line for a game, it also includes a vig. Thus, although a 9 to 4 odds may produce a probability of .69 without a vig, a 10% vig would result in a .76 to .77 probability while a 15% vig is .81 probability.

3. Brian Burke says:

j holz-No, he used the method to suggest a possible mismatch between the spread and over/under point total, and the tradesports win probability. But he did raise the possibility that the method may not hold when one team is a strong favorite.

jaginma- You're right. My bust. And that would make almost everything I wrote wrong.

But assuming the vig is equal for both bets (the 0.60 to win the SB, and the 0.69 to win the AFC), wouldn't they would cancel out in the math:

(0.60*vig)/(0.69*vig) = (0.60/0.69) = 0.87 for the SB individual game

4. jaginma says:

Actually, although you forgot to consider the vig, your assumptions are correct and maybe even strengthened more when the vig is included.

There is usually a high price (vig) to pay when looking at future odds (probabilities) from the wise guys. Look at the current odds to win the AFC for the 4 remaining teams. (Odds are from vegasinsider.com)

NE - 1 to 8
IND - 5 to 2
SD - 8 to 1
JAX - 10 to 1

These translate into probabilities of .89, .29, .11, and .10 respectively.
The sum of these probabilities is 1.39 from which one can only conclude that the vig for this set is 28% -> (1.39 - 1)/1.39
Decreasing the probabilities by 28% would result in adjusted probabilities of .69, .21, .08, and .07 if the vig was distributed evenly for each team.
However, I believe, as you suggested, that there is an over reach of play on the Pats and therefore more of the vig is probably slanted towards them - less for others. The increased vig on the Pats would adjust their probability to win down further. Similiarly, the vig for the NFC is 22%.