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 Impossible Puzzle

I'm leaving this one here not necessarily as a puzzle for you to solve, but as an insight in how freaky these math puzzles can get. This particular one is called ‘The Impossible Puzzle’. Solving it is in fact possible, but that title might hint at how difficult that is. Nevertheless, see how far you can get!

Sam and Priya are two very talented mathematicians.
Their friend Anna approaches them and says: “I have chosen two whole numbers, labeled A and B. Note that A is greater than 1 and B is greater than A. The sum of these numbers does not exceed 100.”

The others nod, so she continues: “In a moment, I will inform Sam of only the sum (A+B), and I will inform Priya of only the product (A×B). These announcements remain private!”
She does so, and the following conversation then takes place:

Priya says, “I don't know the values of A and B.”
Sam responds, “I already knew that.”
“In that case, I do know what their values are,” says Priya.
“Really?”, Sam ponders. “Then so do I.”

Now you too, can know what numbers A and B are.

The Survey

A census taker rings the doorbell and a woman opens. She informs the census taker that she lives in the house with her three sons. “What are the ages of your boys, please?” the census taker asks.
“When you multiply all their ages, the result is 72,” the woman cryptically informs him.

The census taker has a confused look on his face, so the woman adds: “The sum of their ages is the same as the housenumber of the house next door.”
He finds this rather odd, but walks to the house next door, only to return shortly after. “I still don't know, can you give me another hint?”

“Sure,” the woman says, “my oldest son likes strawberries.”
The census taker nods and writes down the ages. What are they?

The Rickety Bridge

Andrew, Beth, Carol and Daniel are on vacation and have spent the day exploring the mountain ranges in Yellowstone National Park. Having lost track of time, darkness is setting in and they are in a hurry to get back. But then there is a rickety old wooden bridge on their path, suspended high over a deep ravine. There's a warning-sign stating that the bridge will only be able to carry the weight of two persons at a time.

No-one is willing to cross the dangerous bridge without the light of a flashlight... unfortunately the group only has one of those with them. They can't risk throwing it, thus it needs to be carried back and forth.

Because of their different ages and fitness levels they will all cross at different speeds. Andrew can cross in 1 minute, Beth in 2 minutes, Carol in 4 minutes and Daniel in 5 minutes. For each duo, the slowest will of course determine the duration of crossing.

Soon it will be night and pitch black. Therefore the group wants to cross the bridge in the minimum time possible. Andrew thinks for a moment and then announces it can be done in 12 minutes. No trick. How?

The Pigeonhole Principle

The Pigeonhole Principle states: If 10 pigeons are put into 9 pigeonholes, then at least one pigeonhole must contain 2 pigeons.

...Okay, I suppose it is not a very difficult principle. So let's use it in scenarios that are a bit harder to grasp right off the bat.

Your drawer contains 10 red socks and 10 black socks. You just woke up, barely got your eyes open, just reaching inside the drawer without looking. What is the fewest number of socks you need to take out to be sure to end up with a matching pair?
The answer is: three socks (← click to see). In this example there are 2 pigeonholes/sock colors (red and black) so with 3 pigeons/socks one of those must always contain a pair. Still with me?

Taking it up a notch then. Your drawer contains 2 red, 8 black, 10 blue and 16 white socks. Once again, what is the fewest number of socks you need to take out to be sure to end up with a matching pair?
Five socks. Think of the pigeons!

I guess that still makes sense. How about this: There must be at least two people living in Rome with the exact same number of hairs on their head. Even if the bald don't count. And that's not a probability either, that's a fact. How do we get to make such a claim?

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!