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47 problems
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1
Exactly Three Heads in Five Flips A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? Express as a fraction in lowest terms.
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Medium2Exactly Two Heads in Five Flips A fair coin is flipped 5 times. What is the probability of getting exactly 2 heads? Express as a fraction in lowest terms.
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Medium3Coin Flip Streak You flip a fair coin 4 times. What is the probability that you get at least one heads? Express your answer as a fraction in lowest terms (e.g. 15/16 ).
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Easy4Two Dice Sum to Seven Two fair six-sided dice are rolled. What is the probability that their sum equals 7 ? Express your answer as a fraction in lowest terms.
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Easy5Inclusion-Exclusion with Three Events In a probability space, P(A) = 0.5, P(B) = 0.4, P(C) = 0.3, P(A∩B) = 0.2, P(A∩C) = 0.15, P(B∩C) = 0.1, and P(A∩B∩C) = 0.05. Find P(A ∪ B ∪ C). Express as a decimal.
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Diamond6Union of Two Events P(A) = 0.6, P(B) = 0.3, P(A ∩ B) = 0.1. Find P(A ∪ B). Express as a decimal.
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Easy7Blackjack Basics In Blackjack, what is the probability of being dealt a natural blackjack (an Ace and a 10-value card) from a single 52-card deck?
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Easy8Pólya Urn Model An urn starts with 1 red and 1 blue ball. You draw a ball uniformly at random, note its color, then return it along with one additional ball of the same color.…
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Diamond9Probability Generating Function Let X be the number of heads in 3 fair coin flips. The probability generating function is G(s) = E[s^X]. Find G(2) , i.e. evaluate the PGF at s = 2. Express as a fraction in lowest terms.
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Diamond10Birthday Pair Probability In a room of 23 people, what is the approximate probability (to the nearest percent) that at least two share the same birthday? Assume 365 days in a year and all birthdays equally likely.
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Medium11Burnside's Lemma: Bead Necklace How many distinct necklaces can be made using 4 beads, each colored with one of 2 colors? (Two necklaces are the same if one can be rotated to match the other. Reflections are considered distinct.)
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Diamond12Medical Test False Positive A disease affects 1% of a population. A test correctly identifies a diseased person 99% of the time (sensitivity) and correctly identifies a healthy person 95% of the time (specificity).…
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Medium13Conditional: Sum Given Even Two fair dice are rolled. Given that the sum is even , what is the probability that the sum equals 8? Express as a fraction in lowest terms.
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Medium14Coupon Collector: 4 Types A cereal box contains one of 4 equally likely toy types. On average, how many boxes must you buy to collect all 4 types? Express as a fraction in lowest terms.
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Hard15Expected Value of a Die Roll You roll a fair six-sided die and win dollars equal to the number shown. What is your expected winnings? Express as a fraction.
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