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Games in Hand and the Conjunction Fallacy

Wikipedia cites this famous logical illusion as the best illustration of what cognitive scientists call "The Conjunction Fallacy."

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

  1. Linda is a bank teller.
  2. Linda is a bank teller and is active in the feminist movement.

The immediate and obvious answer is answer number two. It's apparent that Linda is active in the feminist movement; any answer that includes that detail must be more probable than one that doesn't include it. Just read the description!

The immediate and obvious answer is also spectacularly and demonstrably incorrect. Even if we assign a very high probability of Linda being a feminist, and a very low possibility of Linda being a bank teller, option two can never be more probable than option one. In fact, this holds for every conceivable probability we could attach to these circumstances; the closest they can come is being equally unlikely, if the probability of Linda being a bank teller is zero, or the probability of her being a feminist is one.

Why? For the most intuitive explanation, imagine that the probability of Linda being a bank teller is one, or in other words, we know for a fact that Linda is a bank teller. We can then simply ignore the bank teller condition and focus on the other Is it then more likely that Linda is a feminist, or that Linda simply exists?

More mathematically, probabilities by definition range from between zero and one. To calculate the chances of two events occurring, you multiply the probability of one by the probability of the other happening. Multiplying something by a number ranging between 0 and 1 always results in that number being less. If the probability of Linda being a bank teller is 5% (P = .05) and the probability of Linda being a feminist is 95% (P = .95), then the probability of option 1 being true is simply .05, while the probability of option 2 is .05*.95, or .0475.

It turns out that human brains are really, really bad at evaluating the probability of events with different sample sizes. Our intuition completely lets us down when we're trying to compare the probability of a single thing happening to two things happening in conjunction. Or when we're trying to compare the probability of two events happening compared to four.

Which brings us to games in hand.

Consider the following two potential schedules for one of the leaders in the race for the Shield, Real Salt Lake. One is the real one, one has two more games added.

@Portland, v.Chivas.

@Portland, v.Bayern, v.Chivas, @Barcelona.

Which one of those schedules looks easier to you? The intuitive mind says option one, mostly because the average difficulty of each of those games is significantly lower than the average difficulty of option two. But, much like how adding a conjunctive condition in Linda's case can only decrease probability, adding more opportunities to rack up points only makes it more likely that Real Salt Lake gets to, say, 6 points in their remaining schedule. Even if their odds of stealing a point or three from Bayern or Barcelona are minute, it's still better than nothing.

I imagine we'd be having a different conversation about Seattle's chance for the shield if we had the same number of points accrued and our schedule looked like this:

v.Vancouver, @Dallas

rather than what it does look like, which is this:

v.Vancouver, @Portland, @Dallas, v.Galaxy

But the odd thing is, I think there's a good chance more people would be tapping us to win the shield in option one as opposed to option two, all thanks to the conjunction fallacy.

Obviously real life never ends up lining up so neatly, and the conclusion is rarely as clear as they are in these thought experiments. Still, these cognitive distortions are worth keeping an eye on, and adjusting for, if only in an ad hoc, archer-like way. There's still a good argument to be made that Seattle shouldn't be the favorites for the shield, but we should be on the lookout for where our brain's limitations may blind us.

FanPosts only represent the opinions of the poster, not of Sounder at Heart.

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