# GeekDad Puzzle of the Week: When Are the Odds Even?

This week’s GeekDad Puzzle of the Week is about simple probability, and came up when Max and I were shooting baskets with crumpled up sheets of paper. Basically, the conversation went that if there were two outcomes (making the basket or missing the basket) then there is an equal chance of each, right?

After walking my way through some of the math, I started wondering about situations where there really are even odds. If we have a bag containing equal numbers of black and white marbles, and we pull out one marble, the odds of it being black are even. If we have a bag containing 120 marbles, 85 of which are black, the odds of us pulling out two marbles and them both being black is also even — (85/120)x(84/119) = 0.5 or 50%.

If the largest bag we have can hold 1,000,000 marbles, for how many sets of marbles (i.e., the 120 marbles described above are one set) can we pull two marbles and have a 50% chance of them being the same? Are there any sets of marbles for which we can pull three marbles and have a 50% chance of them being the same? If so, how many?