One of the most famous is the birthday problem. How many people must be in a room before there is a good chance that at least two of them share the day of the year that is their birthday?
Here is a table from my favorite math book, "Finite Math" by Kemeny, Snell and Thompson
Most people guess way too many because they tend to think the question is about a GIVEN person having the same day as someone else. It usually helps to think of the difficulty of adding people to the group while steadily avoiding adding someone who has the same birthday as someone already in the group.. It is similar to the feeling in a bingo group that the numbers called are going to all fall on somebody's card.
Another famous probability problem is called the Monte Hall problem, after the MC on a tv show. Monte has three doors, one of which has a terrific prize behind it. He asks you to pick a door and you do. Then, Monte opens one of the two remaining doors and yep, no prize there. Now it is your turn: ask Monte to open the door you picked or switch to the one he didn't open. The answer is that it is always a better idea to switch. True, you will not get the prize if you happened to pick the prize door at the start. But since Monte has eliminated one of the losing doors for you, you have a much better chance of getting the prize is you switch from your original choice. Doesn't seem right? Didn't to many probability experts, either, but it is proved now. It helps to think of not three doors but 1,000. If you pick a door and Monte opens 998 doors, showing they don't have the prize, do you want to risk opening your original choice or switch to the door he skipped opening?
A better explanation is that there are three doors so you have a ⅓ chance of picking the prize door. That means there is a ⅔ chance the prize is behind one of the other doors and Monte shows you which one that is. So, switching gives you twice the chance for the prize. You can look up "Monte Hall problem" for further discussion.
Bill
Main blog: Fear, Fun and Filoz
Main web site: Kirbyvariety
