The Three Strange Statistics

Here are three statistics that will make you scratch your head.

• Of all people that ever lived, 6.7% are still alive today.
• Whenever you give a deck of cards a proper shuffle, it is safe to say that the order that comes out has never been dealt before in all of human history.
• Statistically, people who get injured will more often be living in an odd-numbered house than an even-numbered house.

Explanations after the jump!

The Prosecutor's Fallacy

The field of statistics has quite a few types of fallacies up its sleeve. The prosecutor's fallacy is a particularly misleading one. In the courtroom, the prosecutor may not purposely use the fallacy to present evidence. Neither may the defense, who might use it to argue a suspects innocence. Still, sometimes the fallacy is presented by mistake.

Let's say some DNA is found on a murder scene. The police have a DNA databank containing 20,000 people and run the sample through it. There is a match, the suspected murderer is identified and put to trial.
The crime scene analyst testifies that the probability of two DNA profiles matching erroneously is only 1 in 10,000. The jury concludes that this means that there is only a 0,01% chance that the suspect is innocent.

Would you conclude the same?

The Sleep Experiment

To make a quick buck, you volunteer for a strange experiment. The details of the experiment are explained to you before you start.

You arrive on a Sunday and are put to sleep. A fair coin is tossed to decide what happens next.

• If the coin comes up heads, you're awakened on Monday only and the experiment ends there.
• If the coin comes up tails, you're awakened on Monday, put back to sleep with a pill, and awakened again on Tuesday. That pill also erases the memory of your last awakening.

This means that whenever you are woken, you do not know what day it is. And each time, the researchers ask you the same question:
“What do you now say is the probability that the coin landed heads?”

So if you are partaking in the experiment and being awakened with that question... what is your answer?

The Necktie Paradox

Tom and Michael are both given a cheap necktie by their respective wives for Christmas. At the office Christmas party they start arguing over which (who) is the cheapest. Having had a few drinks, they agree to have a silly bet. They will ask their wives how much their neckties cost. The guy with the more expensive necktie has to give it to the other as the prize.

Tom eagerly accepts the bet. He reasons that winning and losing are equally likely. “If I lose, then I lose the value of my necktie. But if I win, then I win more than the value of my necktie. Therefore this bet is ultimately to my advantage!”
Michael is eager to accept the bet as well... since he reasons in exactly the same way.

Either guy's reasoning seems sound, yet they cannot both have the advantage in the bet! Where is the fault?

Details: For a fair bet you'd expect both guys to have a 50% chance of winning it (100% together). In the extreme case where one guy would have a 100% chance of winning, it can only follow that the other guy has 0% chance, i.e. no chance at all. So that's why they can't both have the advantage (= a chance bigger than 50%).

The Monty Hall Problem

Suppose you're on a game show, and you are given the choice of three doors: behind one door is a car; behind the other two are goats. Naturally your goal is to win the car, which is equally likely to be behind each door.

You pick a door (e.g. #1), and the host (who knows what's behind each door) opens another (e.g. #3) which has a goat. This means there are two doors left, one with a car and one with a goat. You are now given the opportunity to switch your choice (from #1 to #2). Is it to your advantage to do so?

Since you can't know which of the two remaining doors has the car, and since your initial pick had a chance of one-third, you might think that it does not matter. The chance is still ¹/₃ and you might as well stay with your original choice, right? Wrong! Switching actually doubles your chances to ²/₃.

The Birthday Paradox

Given a group of randomly chosen people, how big is the chance that two persons in this group share the same birthday? You can assume that each day of the year is equally probable for a birthday.

The answer might surprise you. In case of a group size of 23 people, the chance that there are two people in there with the same birthday, is already 50/50. A probability of 99% is reached with a group of only 57 people!