Counter-intuitive statistics puzzles

[NATURE.M]

Recently, I came upon the Monty Hall Problem, and found it to be quite interesting. Any suggestions of similar counter-intuitive statistics puzzles would be much appreciated.

 

[I_am_learning]

Suppose a drug test is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. If a randomly selected individual tests positive, what is the probability he or she is a user?

The answer is not close to 99% as one might think. In fact, it depends on how many other people are drug user? If only 0.5% of people in the group was a user, then even if someone tests positive, the chance of that person being a user is only around 33%.
Details here

 

[micromass]

Puzzle 1:
In the land of Tyrannia, parents want to have boys rather than girls. So the rules of Tyrannia state that everybody who gets a girl, can have another child, until they have a boy. So for example, a couple can get a boy right away, and is then not allowed another child. But a couple can get 10 girls and then a boy, and then is not allowed another child.

What is the eventual percentage of boys and girls?

Puzzle 2:
A couple has two children. One of the children is a boy. What is the probability that the other is a girl.

A couple has two children. The oldest of the children is a boy. What is the probability that the other is a girl.

Are these two probabilities the same?

Other things to review: http://en.wikipedia.org/wiki/Simpson's_paradox

 

[micromass]

How many random people do you need to gather in room in order for the probability that two have the same birthday to be 90% or higher?

 

[micromass]

In order to get a certain tennis award, you need to play three matches. You can play against an easy player or against a very tough player. But you can't play the same player in a row.

So you have the following two options:
Match 1: Play against the easy player
Match 2: Play against the hard player
Match 3: Play against the easy player

or

Match 1: Play against the hard player
Match 2: Play against the easy player
Match 3: Play against the hard player

To get the award, you need to win two matches in a row. Which schedule do you choose?

 

[micromass]

You can play the following game:

You first pay a certain sum as entry fee, namely x dollars.

Then you flip a coin. You keep flipping a coin until you hit head. You count the amount of tails you got, call this $n$. You get paid $2^n$ dollars.

For example, if you throw 3 tails in a row, then you get paid $8$ dollars.

How large should the entry fee in order for you to play this game? What if you play this game several times?

 

[NATURE.M]

Thanks a lot for the suggestions micromass and I_am_learning. The puzzles defiantly seem challenging upon first glance. I'll go through each one in more depth tomorrow, as I'm awfully tired at the moment.

 

[Evo]

Going forward nature, you must provide links to the puzzles.

 

[johnqwertyful]

In order to get a certain tennis award, you need to play three matches. You can play against an easy player or against a very tough player. But you can't play the same player in a row.

So you have the following two options:
Match 1: Play against the easy player
Match 2: Play against the hard player
Match 3: Play against the easy player

or

Match 1: Play against the hard player
Match 2: Play against the easy player
Match 3: Play against the hard player

To get the award, you need to win two matches in a row. Which schedule do you choose?

2 makes sense. Given that you're "very likely" to beat the easy player, you're given two chances to beat the hard player vs only 1 chance.

 

[NATURE.M]

2 makes sense. Given that you're "very likely" to beat the easy player, you're given two chances to beat the hard player vs only 1 chance.

Yeah after thinking over that puzzle thoroughly, the answer seemed rather trivial, although the appearance can be somewhat deceiving.