Rolling dice

You and I play a game. We each have 1 die. I win if I roll a 4. You win if you roll a 5. If I go first, what is the probability that I win?

Some quick remarks:

• the game is over when I roll a 4 or when you roll a 5. Until that happens, the game is played ad infinitum.

• the game is played in rounds. At each round, I roll the die, and only then you roll the die. If I get a 4, I win the game and you don’t get to roll your die.

• the probability that I win is the same as the probability that you lose. And vice-versa.

• if you go first, the probability that I win is different (of course).

• note that there are two dice! However, since we roll the dice in a non-simultaneous manner, we actually only need one die. I can borrow yours, or lend you mine.

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Related:

• The Dice Brain Teaser

Hat-tip: Nuclear Phynance

The Determinant Game

Via Alex Gittens’ blog, here’s a cool game:

Let’s play a two-person game. Start with a zero matrix, and at their turn each player changes one of the zero entries to a number in the set . The first player’s goal is to make the determinant positive, while the second’s is to make it negative. Numbers may be used only once.

What is each player’s “optimal” strategy? If both players play “optimal” strategies, who will win?