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FRM Part I · FRM Exam Part I · Simulation and Bootstrapping

A sample has n = 5 observations: 2, 4, 6, 8, 10. A bootstrap resample of size 5 is drawn with replacement. What is the probability that a given resample is identical to the original ordered set, i.e., the draws in order are 2, 4, 6, 8, 10 exactly, and what is the probability that a resample contains the observation 10 at least once?

The ordered sequence has probability (1/5)^5 = 0.00032. The probability of never drawing 10 is (0.8)^5 = 0.3277, so at least once is 0.6723. The other options confuse missing with including, or use without-replacement logic.

  1. A0.00032 and 0.6723Correct
  2. B0.00032 and 0.3277
  3. C0.0083 and 0.6723
  4. D0.2 and 0.8

Explanation

Each ordered draw has probability 1/5, so the exact ordered sequence has probability (1/5)^5 = 1/3125 = 0.00032. The chance 10 is never drawn is (4/5)^5 = 0.32768, so at least once is 1 − 0.32768 = 0.67232. The 0.3277 option is the probability of missing 10; 0.0083 is 1/5! (sampling without replacement).

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