Expected v. Actual Results in Probability


This post is inspired by a tweet Aubrey sent out, and by this xkcd comic. Aubrey's tweet read:

"If planes had a 50% chance of crashing, would you get on them? #marriage"
This is, of course, a reference to the common claim that half of all marriages end in divorce. I don't have any actual research to confirm or deny that rate, but it's irrelevant to the point of this article. Also, I'm going to go ahead and ignore the fact that comparing a plane crash to divorce is hardly a fair comparison... the point of this article isn't to refute her claim, but to illustrate some math that's often incorrectly used in regards to probabilities.

Basically, there's an assumption at play in Aubrey's tweet (which is what the xkcd comic refers to). The assumption is that a 50% success rate for marriages means that every new marriage has only a 50% chance of succeeding. But this isn't really the case.

Consider a quarter. If you flip it, you have a 50% chance of getting heads. Now, take a quarter, and flip it 10 times. The EXPECTED outcome is 5 heads, 5 tails. But that won't always be the case - you could very easily get 6 or 7 of one (and, in rare instances, 10 of the same). This outcome is the ACTUAL (or experimental) outcome, and it won't always match the expected outcome.

If you flip a coin 10 times, and get 7 heads, you have a 70% success rate (if you call heads a success). But your next flip STILL has only a 50% success rate - the past results have no effect on the future.*

So the fact that half of all marriages in the past failed has no bearing on the success/failure of any one given marriage. And this makes sense, when you think about it: not all marriages are created equally, after all. The odds of any given marriage succeeding depend on a multitude of factors, none of which is the success of all marriages to come before.

The comic plays off this too. The character is referencing the fact that lightning only kills one in 7,000,000 Americans each year, and assumes that makes it pretty safe to play in the lightning. But them being out there drastically increases their likelihood. One in 7,000,000 is an actual result, not the "expected" one.

Long story short - Probabilities determine the results, not the other way around.

*This is the memoryless principle. It's also a key element of the gambler's fallacy - the idea that being on a losing streak means you have a higher chance of winning. That's not true at all - your odds of winning are not affected by the win/loss ratio of the games you've already played.

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