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- Guaranteeing a red ball in draws ●○○○○ 10 ptsA bag contains 10 red balls, 8 white balls and 7 black balls. The balls are drawn one by one at random, without replacement. Find the minimum value of such that a red ball is guaranteed to have been drawn by the -th draw (on the -th draw or earlier).
- Blasting an asteroid into pieces ●○○○○ 10 ptsBruce the Astronaut must destroy a large asteroid. When he blasts a large asteroid, it splits into two medium asteroids. When he blasts a medium asteroid, it splits into three small asteroids. When he blasts a small asteroid, it vanishes. How many blasts does he need to make the asteroid completely vanish?
- Non-leap years between 1801 and 1900 ●○○○○ 10 ptsHow many non-leap years are there from 1801 to 1900 inclusive, in the Gregorian calendar?
- Two heads and two tails ●○○○○ 10 ptsFour fair coins are tossed simultaneously. What is the probability that exactly two coins show heads and two show tails? Give your answer as a decimal.
- Pigeonhole for shared birthdays ●○○○○ 10 ptsWhat is the smallest number of people needed to be certain that at least two of them have the same birthday? Assume there are 366 possible birthdays (including February 29).
- Choosing two balls from twenty ●○○○○ 10 ptsTwenty balls numbered 1 through 20 are in a bin. In how many ways can 2 balls be selected if the order in which they are drawn does not matter?
- Matches in a knockout tournament ●●○○○ 20 ptsThere are 100 nations taking part in an elimination tournament, in which a team is eliminated as soon as it loses a match. How many matches must be played in total to determine the overall winner?
- Maximum months with five Sundays ●●○○○ 20 ptsA month can have at most 5 Sundays. In a year of 12 months, what is the maximum number of months that can have 5 Sundays?
- Aces before the joker ●●○○○ 20 ptsA standard 52-card deck plus one joker (53 cards in total) is shuffled into a uniformly random order and turned over one card at a time. What is the probability that all 4 aces are turned over before the joker? Give your answer as a decimal.
- Blue points on a line ●●○○○ 20 ptsOn a line, one hundred points are marked, each painted green, blue, or red. Between any two red points there is a blue point, and between any two blue points there is a green point. There are no fewer red points than blue points, and no fewer blue points than green points. How many points are painted blue?