1. A year on planet Ork has 100 days. Find the smallest number of Orkians for which the probability that at least two of them have the same birthday is 50% or more.
2. In many state lotteries, six numbers are selected from a set of numbers. Quite often, the winning selection contains two consecutive numbers. When six numbers are selected from the numbers 1 through n, the probability that there will be two consecutive integers in the selection is .5771 when n is 40 and the probability is .4209 when n is 60. Determine the largest value of n for which the probability of having two consecutive numbers is greater than .5. Note: There are C(n - k + 1, k) ways that k numbers selected from the numbers 1 through n will have no two consecutive numbers.
3. How many times must you roll two dice so that the probability of rolling a pair of 6s at least once is greater than or equal to .5? This question was posed in 1654 by the French writer and gambler Chevalier de Méré?