Odd Weekend

Take a look at lottery ticket sales or the annual revenue of a Las Vegas casino and it becomes apparent that many people have a difficult time wrapping their heads around probability. But even someone like myself, who has a basic understanding of the law of large numbers, can be caught off gaurd.

This weekend I was reminded of a game called Penny-Ante. It took me an hour of thinking and experimenting to finally accept the results and understand the logic.

Penny-Ante

There are two players and one coin. Player one chooses a sequence of three flips (ie Heads-Tails-Heads) and then player two chooses a sequence. The penny is flipped and the results tracked until one of the players’ sequences shows up. That player wins.

There are eight possible sequences: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Each sequence has a 1/8 chance of showing up.

So what’s the trick? Player two has an advantage over player one. He knows the sequence that player one chose before he chooses his own. Say player one chooses HTH. Player two can improve his chances if he takes the first two values chosen by player one (HT) and makes those his last two (?HT). His first value is going to be the opposite of his last. So player two chooses HHT. This formula can change player two’s odds from 1/8 to 2/3, 3/4 or 7/8 depending on which sequence player one chooses.

Don’t get it? Read this.

So I was thinking about Penny-Ante and remembering some other odd things that challenged my worldview.

Approximating Pi with Buffon’s Needle Method

A couple of years ago a friend of mine told me that you can approximate pi by randomly dropping a needle onto a piece of paper. I didn’t really believe him so I asked him to do it.

He requested a needle and a sheet of ruled notebook paper. He cut the needle so that it’s length was exactly the same as the distance between the lines on the notebook paper, then started dropping the needle on the paper. He kept track of how many times the needle landed on a blue line, and how many times it landed in white space.

After fifty needle drops he multiplied the number of needle drops by two and divided that number by how many times the needle landed on a line. He ended up with 3.134. I was shocked he could get so close. He said it would get closer to pi the more drops he made. He continued until he had 100 drops, multiplied the number of drops by two, and divided by the number of times the needle landed on a line. The result was 3.139.

How does randomly dropping a needle on paralell lines give you pi? My friend gave me detailed mathematical explaination that I couldn’t quite follow, but essentially the root of the problem involves calculating the probability that the needle will cross a line. This probability is dependant on the angle of the needle to the parallel lines. The equation for calculating this probabilty includes the constant pi because the needle can fall anywhere from 0-360 degrees. What you do when you drop the needle is work backwards from that equation. More detailed maths can be located here.

Let’s Make a Deal

Here’s one that I stumbled upon this weekend and still haven’t quite wrapped my head around.

You find yourself on a game show called “Let’s Make a Deal.” The host, Monty, has placed one million dollars behind one of three doors. The other two have gag prizes. You are asked to pick a door. You get to keep whatever is behind it.

You pick door number two. Monty then makes the game interesting. He opens one of the other two doors to show you a gag prize, say door number three. He asks if you would like to change which door you picked or stay with door number two. What should you do?

I assumed, like most people, that it does not matter if you keep door number two or change to door number one. The money has not moved so your chances are the same. I was wrong, you should change your pick to door number one.

Don’t ask me to explain, this guy already did and I’m still kind of confused.