Probability math debate

So me and a friend else were having a debate recently. And I thrive to get a good answer to explain who's right.

So it's mainly about doubling someone's chances.

Here's the situation: There are 10 people at a party, and they all place their names (on a piece of paper) in a hat. The host of said party (who is included in the 10 people) is a nasty cheater and wants to double his chances. So he places his name in twice.

My friend claims that he Successfully doubled his chances because everybody else's chances are at 1/11 and his is 2/11, and 2/11 = 1/11 * 2.

But my logic goes as follows: he originally had 1/10 chance. Double that is 2/10. But by adding his name in twice he also increased the amount of names, so his new chance is 2/11. But 2/11 < 2/10, so he increased it by LESS than 2x.

Either my logic is really messed up, or his is. Who's correct.

• is the 1/12 a typo? Please explain where 12 comes from. Aug 8 '15 at 7:02
• @user2740 Yes, fixed it lol Aug 8 '15 at 13:53
• The host's chance is originally 10%.. If your friend is right, then by putting in ten slips, the host can multiply his chance by 10 --- making it 100%. Ask your friend if he believes the host can guarantee himself a win by putting in ten slips. Aug 8 '15 at 13:59

The problem here is really more in semantics than anything else.

In the first case, the host has 1/10 chance of his name being pulled, like everyone else.

In the second case, the host has 2/11 chance in his name being pulled, while everyone else has 1/11 chance of their name being pulled.

While in the first case the host has the same chance as everyone else of winning, in the second game the host has twice as much chance as everyone else of winning. So in some sense, the host did "double" his chance of winning relative to everyone else.

But to say he doubled his original chances would be false, since as you said doubling his original chances would mean he has a 2/10 chance of winning. So he didn't double his original chances, but in the second game he has double the chance of winning as everyone else.

• Thank you, you really cleared it up for me. Aug 8 '15 at 3:46
• @warspyking You're welcome! Aug 8 '15 at 3:47

I would say you're half right. "Doubling his chances" should be relative to the chance he had previously. But $2/11 < 2/10$, so his chance increased less than double. That's the half where you were wrong.

"Doubling your chances" is a phrase similar to "doubling your money." It doesn't matter what chances (or what money) someone else has before or after you double yours; what matters is what you had before, and what you had after.

Consider the scenario where the host puts in one extra slip with his name on it, and $19$ additional slips with the words "everyone loses". Now his chance to win is certainly twice as much as everyone else's--it's $1/15$, whereas they each have a $1/30$ chance--but his chance of winning is actually less than he started out with: $1/15 < 1/10$. I can't speak for anyone else, but if I a have the opportunity to double my chances at something, I want to end up with better chances than I had before I "doubled" them.