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diff --git a/statistics.mime b/statistics.mime index 0a4397f..f7d16bc 100644 --- a/statistics.mime +++ b/statistics.mime @@ -11,15 +11,29 @@ The real problem is that humans seem to have a gut instinct for stats, but that This leads to the tricky situation where a random person will attempt to interpret a statistic, or generate a statistic, and feel like they're probably pretty close. Sometimes someone who even knows that math will second guess it because it just doesn't feel right. +I'm going to attempt, in this article, to cover a collection of cases where I've seen people often have the wrong idea. + == The Birthday Paradox == -Brief specification of the paradox. -Both the setup and the seamingly ridiculous result. +The Birthday Paradox is one of the simplest and easiest examples of how wrong a person's gut instinct can be. +It goes something like this: How many people need to be in a group to have a 50% chance that two or more of them share a birthday? + +So, most people, when presented with this go through a thought process like the following: + Alright, the typical year has 365 days. A person can be born of any of those days evenly. + We want a 50% chance of collision, so I'd guess (365/2) = 182ish. + So, I'd guess about 182. + +Not bad reasoning. + +The real answer is 23. +I'll explain why that is after a covering a little bit of the basics of probability. == Probability Basics == 3 in 5 means that, if you do the experiment a huge number of times, then about 3/5th of that should be the given outcome. +#### TODO: Mention the conversion from odds to probaility, and mention probability being between 0 and 1. + That's all. If you do an something 5 times and don't get 3 of the given outcome, then that doesn't necessarily mean the probability is wrong. @@ -41,11 +55,64 @@ Take, for example, rolls of a fair die: Each side of the die has a 1 in 6 chance. So, the probability of rolling either a 1 or a 2 is (1/6 + 1/6 = 2/6). This makes sense. -The probability of rolling a 1, then a 2 is (1/6 * 1/6 = 1/36). +The probability of rolling a 1, followed by a 2 is (1/6 * 1/6 = 1/36). + +If you want the probability that the opposite of something happens, you just need to subtract it from 1. + +For example, the probability that two dice each come up 1 is (1/36). +The probability that doesn't happen is (1 - (1/36) = 35/36). == The Birthday Paradox Revisited == -Work through of the math showing why the answer is what it is. +So, now that we've got the basics of probability, let's see if we can work out why the answer to the birthday paradox is what it is. + +First off, assumptions. +I'm assuming that people are born with an equal probability on any day of the year. +That's not quite true in practise, there is a clustering in certain areas of the year, but that would make it more likely that people would have the same birthday, not less, so that's acceptable. + +First off, calculating the probability that a group of people all have unique birthdays is easier than computing the probability that they have 1 or more collisions. +Luckily, since "having everyone have a different birthday" and "having everyone not have a unique birthday" are opposite outcomes, we can subtract that probability from 1 and get the value we actually want. + +So, the probability of the first person having a unique birthday is (365/365 = 1). +That makes sense, since there's only one of them. + +The second person has only 364 days to choose from (since it has to be different from the first), which leaves a probability of (364/365). + +The third person has (363/365). + +So, to compute the probability that three people have unique birthdays we have (365 * 364 * 363) / (365 * 365 * 365), which is 0.99 +That's pretty likely. + +So, the probability that there's one or more of them that share a birthday is (1 - 0.99 = 0.01). + +That proabability rises quickly, though, as we add more people. + +|= Number of People |= Probability of Sharing a Birthday | +| 1 | 0 | +| 2 | 0.003 | +| 3 | 0.008 | +| 4 | 0.016 | +| 5 | 0.027 | +| 6 | 0.040 | +| 7 | 0.056 | +| 8 | 0.074 | +| 9 | 0.094 | +| 10 | 0.117 | +| 11 | 0.141 | +| 12 | 0.167 | +| 13 | 0.194 | +| 14 | 0.223 | +| 15 | 0.252 | +| 16 | 0.283 | +| 17 | 0.315 | +| 18 | 0.346 | +| 19 | 0.379 | +| 20 | 0.411 | +| 21 | 0.443 | +| 22 | 0.475 | +| 23 | 0.507 | + +So, we can see that by 15 people we've got approximately a 25% chance that there will be a shared birthday, and by 23 people we've reached 50%. == The Weather ==