Practice Statistical Simulation in Statistics

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

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

Can't calculate the probability mathematically? Simulate it! Run the scenario thousands of times with random numbers and see what fraction of outcomes match your event. It's like conducting experiments without real resources.

Showing a random 20 of 50 problems.

Example 1

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.

Example 2

hard
You simulate a game where each round you toss a coin and win $1 for heads, lose $1 for tails. After 1000 simulated rounds, total winnings are $24. Estimate P(heads)P(\text{heads}).

Example 3

challenge
You simulate 100000 trials and estimate p^=0.205\hat{p}=0.205. Construct an approximate 95% confidence interval using p^±1.96p^(1−p^)/n\hat{p}\pm 1.96\sqrt{\hat{p}(1-\hat{p})/n}.

Example 4

easy
A simulation of 500 trials estimated P=0.62P=0.62. How many trials matched the event?

Example 5

medium
A simulation models a 0.70.7-probability free throw. In 1000 simulated shots, 715 went in. Estimate the make probability and the miss probability.

Example 6

medium
A simulation estimates a probability as 0.50.5 with 100 trials. About how many trials are needed to cut the standard error in half?

Example 7

medium
To estimate the chance a baseball player with a 0.3000.300 average gets at least 2 hits in 4 at-bats, you simulate. In 5000 simulated games, 1410 had 2+ hits. Estimate the probability.

Example 8

medium
You simulate 5000 trials of two dice rolls to estimate P(sum=7)P(\text{sum}=7) and observe 845 matches. Estimate the probability.

Example 9

easy
To simulate rolling a fair 4-sided die using digits 1-8, how would you assign digits to faces?

Example 10

easy
A simulation runs 200 trials and an event occurs 50 times. Estimate its probability.

Example 11

medium
To estimate the area of an irregular shape inside the unit square, you sample 5000 uniform points; 1750 fall inside the shape. Estimate the area.

Example 12

easy
In 5000 simulated coin flips, 2530 landed heads. Estimate P(heads)P(\text{heads}).

Example 13

medium
A multiple-choice test has 10 questions, each with 4 options (one correct). A student guesses randomly on every question. Design a simulation to estimate the probability of passing (getting 6 or more correct).

Example 14

challenge
A simulation of the 'birthday problem' for 23 people (365 days) yields 5078 trials with a match out of 10000. Estimate P(shared birthday)P(\text{shared birthday}) and compare to the theoretical value of about 0.5070.507.

Example 15

easy
To simulate an event with probability 0.30.3 using a random number in [0,1)[0,1), what range counts as success?

Example 16

easy
A spinner has a 14\tfrac14 win region. Using digits 0-9, which digits could represent a win?

Example 17

easy
A simulation of 400 trials estimates P=0.18P=0.18. How many trials produced the event?

Example 18

medium
To simulate the probability that a 5-person group has at least one shared birthday-month (12 months), you run 10000 trials and find 6190 hits. Estimate the probability.

Example 19

medium
To estimate the probability that two people in a group of 5 share a birthday-week (52 weeks), you simulate 10000 groups and 1850 had a match. Estimate the probability.

Example 20

medium
A game wins with probability 0.20.2. You simulate 50 plays using digits 0-9, letting digit 0 or 1 mean a win. Out of 50 digits, 12 were 0 or 1. Estimate the win probability.
← Back to Statistical Simulation Worked Examples