Statistical Simulation Statistics Example 4

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

Example 4

hard
Two players play a game where each rolls a die. If the sum is 7, Player A wins; otherwise Player B wins. Run a conceptual simulation of 20 trials with these random rolls: (4,3), (2,5), (6,1), (3,3), (5,2), (1,4), (6,6), (2,3), (4,4), (5,1), (3,4), (6,2), (1,1), (2,6), (5,5), (4,1), (3,6), (2,2), (6,3), (1,5). Estimate each player's probability of winning.

Solution

  1. 1
    Step 1: Check each pair for sum = 7: (4,3)=7✓, (2,5)=7✓, (6,1)=7✓, (3,3)=6✗, (5,2)=7✓, (1,4)=5✗, (6,6)=12✗, (2,3)=5✗, (4,4)=8✗, (5,1)=6✗, (3,4)=7✓, (6,2)=8✗, (1,1)=2✗, (2,6)=8✗, (5,5)=10✗, (4,1)=5✗, (3,6)=9✗, (2,2)=4✗, (6,3)=9✗, (1,5)=6✗. Player A wins: 5 times. Player B wins: 15 times.
  2. 2
    Step 2: Estimated P(A)=520=0.25P(A) = \frac{5}{20} = 0.25, P(B)=1520=0.75P(B) = \frac{15}{20} = 0.75. Theoretical: P(A)=636=16≈0.167P(A) = \frac{6}{36} = \frac{1}{6} \approx 0.167. The simulation overestimates A's probability due to the small sample size.

Answer

Simulation estimate: P(A)=0.25P(A) = 0.25, P(B)=0.75P(B) = 0.75. Theoretical: P(A)≈0.167P(A) \approx 0.167. The 20-trial simulation differs from theory due to small sample size.
Running a simulation by hand demonstrates the process but also shows the limitation of small sample sizes — estimates can differ significantly from theoretical values. With thousands of trials, the simulation estimate would converge to the theoretical probability.

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 2 medium

A cereal brand puts one of 6 different toy figurines in each box (equally likely). Use simulation to

Example 3 medium

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

← All Statistical Simulation Examples Statistical Simulation Concept