47. The Birthday Scam
class: closer | impressiveness: 4/5 | factors: large groups, possible giant payoff | requires: large group of people | watch full episode
Quick: in your own mind, imagine you’re at a party with 35 to 40 people. What do you think the odds are that two people will share the exact same birthday?
Remember, we’re looking at about 40 people in the room, and there’s 365 days in the year, so would the odds be…
(a) 0-25%
(b) 25%-50%
(c) 50%-75%
(d) 75%-85%
(e) Over 85%
Make your guess right now. (I’m serious. I want you to try this.) I’ve asked this question at hundreds of lectures across colleges nationwide, and the vast majority of people tend to pick (a) or (b), with the occasional smartass going as high as (d)… but the shocking truth is that with 40 people in the room, the answer is (e), there’s over an 85% chance two people in the room have the same birthday.
This seems utterly insane to me! This is a case where the human mind just isn’t wired correctly to grasp true probability. Most of us figure “okay, there’s about 40 people and about 365 days… so the answer must be about 40/365… which is about 10%.” However, the true math is far more complex. To learn more about it, check out this Wikipedia article on the paradox.
Warning: when you visit that article you’re going to see fancy math stuff like this:
…P(A’) is equal to the product of these individual probabilities:
(1) P(A’) = 365/365 × 364/365 × 363/365 × 362/365 × … × 343/365
The terms of equation (1) can be collected to arrive at:
(2) P(A’) = (1/365)23 × (365 × 364 × 363 × … × 343)
Evaluating equation (2) gives P(A’) = 0.492703
Therefore, P(A) = 1 − 0.492703 = 0.507297 (50.7297%)…
…but thankfully, you’ll also see this super-simple graph showing the range of probabilities like this: Here are a few numbers to keep in mind when you’re trying to decide if the time is right for you to play this trick:
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The odds are 50% you’ll have a match with only 23 people in the room
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The odds are 80% you’ll have a match with only 35 people in the room
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The odds are 90% you’ll have a match with only 42 people in the room
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The odds are 99% you’ll have a match with only 57 people in the room (wow!)
With these numbers in mind, make sure to adjust your wager accordingly, but definitely remember that any time there are more than 23 people in the room, it’s worth it for you to offer a simple even-money proposition.