Solve Graduate Trader Probability Questions

A graduate trader online assessment reportedly included the following quantitative reasoning, probability, and logic questions. Solve each problem. 1. How many distinct arrangements can be formed from the letters in the word `MISSISSIPPI`? 2. Three numbers have pairwise sums 39, 48, and 51. What is the largest of the three numbers? 3. A bag contains 3 green cubes, 4 red cubes, and 2 blue cubes. If 3 cubes are chosen uniformly at random without replacement, what is the probability that all 3 cubes are the same color? 4. Using the same bag, what is the probability that the 3 chosen cubes contain exactly 2 distinct colors? 5. In a horse race with 5 horses, assume every finishing order is equally likely. Which option has the highest expected value? - (a) Pay $10 to bet the exact order of all 5 horses; payout $1000 if correct. - (b) Pay $20 to bet the exact order of the first 4 horses; payout $1000 if correct. - (c) Pay $30 to bet the exact order of the first 3 horses; payout $1000 if correct. - (d) Pay $60 to bet the exact order of the first 2 horses; payout $1000 if correct. - (e) Walk away with $10. 6. Jim flips a fair coin until he gets two consecutive equal results for the first time, either `HH` or `TT`. What is the probability that he flips the coin an even number of times? 7. Antony collects toys from cereal boxes. There are 5 equally likely toy types, one per box, and each box costs £4. What is the expected total cost to collect all 5 toy types? 8. You roll a fair six-sided die. Whenever the result is greater than 3, you roll again. Your payout is the sum of all rolls. What is the expected payout? 9. A disease affects 1% of the population. A test returns positive 99% of the time for someone with the disease, and 5% of the time for someone without the disease. Given that a person tests positive, what is the approximate probability that they actually have the disease? 10. Three fair six-sided dice are rolled. What is the probability that the results are in strictly increasing order? 11. Three fair six-sided dice are rolled. What is the probability that the product of the three results begins with the digit 1? 12. How many rectangles, including squares, are contained in an 8 × 8 chessboard? 13. Alice and Bob each flip a fair coin until they get their first heads. What is the expected number of flips Alice takes, given that Alice finishes in fewer flips than Bob? 14. Four people appear in order: a knight, a liar, a jester, and a child, one role each. - The knight always tells the truth. - The liar always lies. - The jester tells the truth if and only if the previous speaker lied. - The child tells the truth if and only if the previous speaker told the truth. Their statements are: - Person 1: "I am a jester." - Person 2: "I am not a knight." - Person 3: "I am the knight." - Person 4: "I am not a liar." How many of the four statements are true? 15. Three distinct integers are chosen uniformly at random from 1 through 30. What is the probability that one of the chosen numbers is the average of the other two? 16. One box is chosen uniformly at random from three boxes: - Box 1 contains 20 chocolate cookies. - Box 2 contains 10 oat cookies. - Box 3 contains 5 chocolate, 5 oat, and 5 raspberry cookies. Two cookies are then drawn without replacement from the chosen box, and both are chocolate. What is the probability that the chosen box was Box 1? 17. A very large bag contains balls in the proportions 1/2 white, 1/3 red, and 1/6 blue. Balls are drawn without replacement. Given that the first ball drawn is white, what is the approximate probability that the second ball is also white?