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#72 Match up

Set by Christopher Dearlove

Consider a game where two players, A and B, play a sequence of rounds. After each round, one of them wins a point. Player A is the better player, winning 2/3 of all rounds, while player B only wins 1/3 of all rounds, all rounds being independent. The game is won by the first player to have won two more points than the other.

Starting level, after two rounds, what is the probability of player A having won the game?

What is the probability of player A eventually winning the game? (Hint: After two rounds, either player A has won, player B has won or the players are back to level.)

If, at some point in the game, player B is one point up, what is the probability that player B will win the game?

Solution next week

#71 Dip die

Solution

You need 5 × 5 × 5 = 125 cubes. Of these, the central 3 × 3 × 3 cube of cubes will be unpainted, so 125 – 27 = 98 cubes will have some paint on.

There are 125 × 6 = 750 total faces, of which 25 × 6 = 150 are on the outside of the cube. Since picking a cube and rolling it is equivalent to picking a cube face at random, the probability of a red face is 150/750 = 1/5.

This can also be seen by observing that if the cubes are laid out with each in its original orientation, 1/5 of the faces pointing in each direction are red.