Puzzle no 1
At a particular school, an enchanted hat assigns each incoming student to one of four residential houses. School lore says that the hat sorts students into houses entirely on the basis of personality and moral character. But this year, a rumor is circulating that the hat actually assigns each student completely randomly.
The Headmaster hopes to disprove the rumor and reasons that if she can show the hat's assignments to be somewhat predictable, it cannot be assigning randomly. Before the students are sorted, she speaks with each student and privately writes down her prediction of which house that student will be assigned to.
If she does not predict accurately every time, what is the smallest number of students (m), and the corresponding number of correct predictions (n, n < m) that would give us a high level of confidence (say, 99% certainty) that the hat is not completely random in its sorting?
If the headmaster's predictions are always wrong, would a sufficient number of wrong predictions (w) disprove the rumor with similarly high confidence?
Puzzle no 2
You have 5 unmarked bags with 100 beads each. Bags #1-4 contain 4 red beads and 96 black beads; bag #5 contains 7 red beads and 93 black beads. You randomly select one of the five bags and remove three beads without looking inside the bag. One is red, and the other two are black. What is the probability that you drew the beads from bag #5?
Return the three beads to the bag and give it a good shake to mix things up. In the best-case scenario, what is the minimum number of beads you can withdraw, one at a time, to identify this bag as bag #5 with at least 50% certainty?