How to solve Likelihood-list questions

Rank three events from most to least likely. Fix the denominator first, then check subsets. Press 1 to 6 to choose a ranking.

How to solve them

1. Cue: "rank from most to least likely". Do not compute everything exactly. Compare fractions.
2. Step 1: fix the denominator. "Given that ..." or "a random person from ..." shrinks the world, so the denominator is the size of that group, not the total.
3. Step 2: check subsets. If one event is inside another, it cannot be more likely. "Red and a face card" is never more likely than "a face card".
4. Step 3: compare fractions by cross-multiplying, or convert to decimals with the anchors (1/3 = 0.333, 1/6 = 0.167, 1/7 = 0.143, 3/8 = 0.375). Commit within seconds.

Worked examples

• Rank these three events from most likely to least likely.
  A. You roll two fair dice. Given that the sum is at least 9, the sum is a multiple of 3.
  B. You roll two fair dice. The product is at least 12.
  C. You roll two fair dice. The sum is at least 9.

  1. A > B > C
  2. A > C > B
  3. B > A > C
  4. B > C > A
  5. C > A > B
  6. C > B > A
  Answer: A > B > C

  1. A: 5/10 = 0.500
  2. B: 17/36 = 0.472
  3. C: 10/36 = 0.278
  4. Ranked from most to least likely: A > B > C.

• Rank these three events from most likely to least likely.
  A. A random person with the result "Pass" is in "Students".
  B. A random person from "Workers" has the result "Pass".
  C. A random person from "Students" has the result "Pass".

  	Pass	Fail
  Students	53	12
  Workers	45	26
  Retirees	50	21

  1. A > B > C
  2. A > C > B
  3. B > A > C
  4. B > C > A
  5. C > A > B
  6. C > B > A
  Answer: C > B > A

  1. A: 53/148 = 0.358
  2. B: 45/71 = 0.634
  3. C: 53/65 = 0.815
  4. Ranked from most to least likely: C > B > A.

• Rank these three events from most likely to least likely.
  A. You draw one card from a standard deck. Given that it is a face card, it is red.
  B. You draw one card from a standard deck. Given that it is red, it is a face card.
  C. You draw one card from a standard deck. It is a heart.

  1. A > B > C
  2. A > C > B
  3. B > A > C
  4. B > C > A
  5. C > A > B
  6. C > B > A
  Answer: A > C > B

  1. A: 6/12 = 0.500
  2. B: 6/26 = 0.231
  3. C: 13/52 = 0.250
  4. Ranked from most to least likely: A > C > B.

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Optiver · Da Vinci

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