FRM Part I · FRM Exam Part I · Simulation and Bootstrapping
An analyst bootstraps a sample of 5 returns: 1, 2, 3, 4, 5 (in %). One bootstrap resample is drawn with replacement, size 5. What is the probability that the resample contains the value 5 at least once?
The probability is 1 minus (4/5) to the fifth power, which is about 67.2%. Each draw misses the value 5 with probability 0.8, so all five draws missing has probability 0.32768, and the complement gives the chance of at least one appearance.
- A1 - (4/5)^5 = 67.2%Correct
- B1 - (1/5)^5 = 99.97%
- C5 x (1/5) = 100%
- D(1/5)^5 = 0.03%
Explanation
Each draw misses the value 5 with probability 4/5. All five draws miss with probability (0.8)^5 = 0.32768. So the chance of at least one 5 is 1 - 0.32768 = 0.67232, or about 67.2%. Adding probabilities (option 5 x 1/5) double counts overlapping outcomes.
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