Hi Everybody!

After being gone for a while, I'm back here with a quick edition of The Nation Has Problems for pledge week. I didn't have a lot of time to put things together, but I came up with something that should be pretty fun! Probability questions are always good for people that are interested in sports, so I've got three questions of varying difficulties to attempt!

Easy

Suppose you roll three dice. What is the probability that you roll at least one 5 or 6?

Medium

Two players play a game involving flipping coins. Each player has a particular sequence of Heads and Tails that they'd like to see. Once one of the two sequences occur, then that player wins and the game is over. For instance, suppose Player 1 has HHH and Player 2 has TTT, then the following sequence:

HHTHTTHHTTHTHTTHHH

In this case, Player 1 is the winner.

Here are three different pairs of sequences for the players. In each case, what is each player's chance of winning?


a) Player 1: HHH, Player 2: TTT
b) Player 1: HHH, Player 2: TT
c) Player 1: HTT, Player 2: TTH

Hard

A banquet is thrown for 100 current and ex-Twins players. Each player is assigned his own seat. Unfortunately, A.J. Pierzynski is the first player to enter and he completely ignores the assigned seating and just sits in the seat closest to him. Each other player that comes in after him then sits in his own seat if it is available and otherwise picks a seat at random. Joe Mauer is the last player to enter. What is the probability that Joe sits in his assigned seat?

Good luck and have fun! --GH