Catching up on some answers - a few days ago, I presented this puzzle:

A constant flow rate pump is hooked to a balloon. At t = 0, the diameter of the balloon is 1 foot. The pump will run for a random time inflating the balloon, but a switch will stop it if the diameter gets to be 3 feet. Assume the tensile strength of the balloon is sufficiently small that no pressure difference exists between the inside of the balloon and the atmosphere. The game is to bet on the range of the final diameter. You can bet it is between 1 and 2 feet or between 2 and 3 feet. The odds offered are even. How do you bet?

The catch here is the possibility of confusing parameters. The volume increases linearly with time, but the diameter increase with the cube root of the volume. This means that more time will be spent going from 2 to 3 than was spent going from 1 to 2. Therefore you should bet on 2 to 3 and you have a definite advantage.

Then I presented a variation of a card game:

A deck of 52 cards is shuffled and placed face down. You place a bet and you are allowed to let any number of cards go by without being turned up. When you say stop, you are betting the next one turned up will be a black ace. Your payoff is based on the number of cards left in the stack. For instance, if there are only 4 cards left, and two of them could be aces, then your payoff is even. If you chose the first card, then the payoff is 25 to 1 and so on for the other positions. Given this modification, what is the most favorable place for you to bet? You can take any card from the top to the third from the bottom.

The catch here was to lure you into thinking the probability of picking the ace changes as the stack is lowered, but if you have not seen any of the cards, the odds of winning remain the same, but since the payoff changes as cards are turned, then you should obviously take the first card since if you win, you get a lot of money, but with the same chance of winning, your payoff decreases with each card turned. This is another example of misdirection to confuse the mark.

Then I proposed a final variation. This time the cards are turned face up one at a time until you say to stop. Then you are betting the next card will be a black ace. What is your best strategy now? How does the game change if you are allowed to keep going down to the last card? Here the chances of turning up an ace on the next card does increase as the stack is lowered, providing you have not already found the ace. Since you lose if it comes up and you have not chosen, you must balance the risk against the increasing chance of winning. The analysis of this version is more involved, and several people seem to be still working on it, so I will wait for a while before discussing it in more detail. The point of this variation was to demonstrate the changes that happen when you have more information by seeing the rejected cards. This is really very different from the previous variation.

Finally, here is an elementary problem having nothing to do with probability that I was reminded of when making a left turn in a multiple lane turning chute. Imagine you have a rope wrapped around the equator of the earth. It is an ideal rope and just fits snugly around an ideally smooth and spherical earth. You cut the rope and insert a one-foot section. If the expanded rope is raised uniformly above the surface of the earth, how high does it get? Can you shove a sheet of paper under it, or drive a truck under it? How high?

This is related to the distance saved by taking the inside lane on a turn. Assume we have two 90 degree turns. Both have two lanes. The mean turning radius of the inside lane is 50 feet for the first one and 2,500 feet for the second one. What is the difference in path lengths, inside lane to outside lane, for each case. Assume normal width lanes that are immediately adjacent to each other.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.