Statistical Simulation Statistics Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
A cereal brand puts one of 6 different toy figurines in each box (equally likely). Use simulation to estimate how many boxes you need to buy to collect all 6 figurines.

Solution

  1. 1
    Step 1: Model: generate random integers from 1 to 6. Each number represents a figurine. Keep generating until all 6 numbers (1–6) have appeared at least once. Record the count of boxes needed.
  2. 2
    Step 2: Repeat this simulation many times (e.g., 500 trials) and record the number of boxes needed each time.
  3. 3
    Step 3: Average the results across all trials. The theoretical expected value (from the coupon collector's problem) is 6×(1+12+13+14+15+16)=6×2.45=14.76 \times (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}) = 6 \times 2.45 = 14.7 boxes.

Answer

Simulate by generating random numbers 1–6 until all 6 appear, counting boxes each trial. Average over many trials. Expected result: approximately 14.7 boxes.
Simulation is powerful for problems that are hard to solve analytically. The coupon collector's problem has a theoretical solution, but simulation provides an intuitive way to estimate the answer and understand the variability in outcomes.

About Statistical Simulation

Using random number generation to model real-world processes and estimate probabilities or outcomes that are difficult to calculate theoretically.

Learn more about Statistical Simulation →

More Statistical Simulation Examples

Example 1 easy

You want to estimate the probability of getting exactly 2 heads when flipping 3 coins. Describe how

Example 3 medium

A multiple-choice test has 10 questions, each with 4 options (one correct). A student guesses random

Example 4 hard

Two players play a game where each rolls a die. If the sum is 7, Player A wins; otherwise Player B w

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