Dice probability

Question

How many times do you have to roll two six-sided dice for there to be at least 50% chance that you roll two 3's at least once?

Answer 1

This means that P(0 pairs of 3s in n rolls) < 1/2

P(0 pairs of 3s in n rolls) = P(all non-pairs of 3s in n rolls)

As the P(pair of 3s) = 1/6 * 1/6 = 1/36, P(non-pair of 3s) = 1 - 1/36 = 35/36

P(all non-pairs of 3s in n rolls) = (35/36)^n

(35/36) ^ n < 1/2

Taking logs

n ln (35/36) < ln(1/2)

n ln(36/35) > ln(2)

n > ln(2)/ln(36/35)

n > 24.6051

As n is an integer, n = 25