Math 2 Quiz: Simulating Probabilities
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Simulating ProbabilitiesQuestion 1 of 20

A lottery simulation uses the following method: generate 6 distinct random integers from 1 to 49, then check if they match a predetermined winning combination. After 100,000 trials, the simulation estimates the probability of winning as 0.0000089. The theoretical probability is 1C(49,6)≈7.15×10−8\frac{1}{C(49,6)} \approx 7.15 \times 10^{-8}. What is the most likely explanation for the discrepancy?

The sample size of 100,000 is insufficient to accurately estimate such a small probability
The theoretical calculation is incorrect because it doesn't account for the order of selection
The random number generator is not producing truly uniform distributions across the range 1-49
The simulation contains a coding error that makes winning more likely than it should be
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Math 2 Quiz

Math 2 Quiz: Simulating Probabilities

Practice Simulating Probabilities in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Simulating Probabilities, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A lottery simulation uses the following method: generate 6 distinct random integers from 1 to 49, then check if they match a predetermined winning combination. After 100,000 trials, the simulation estimates the probability of winning as 0.0000089. The theoretical probability is 1C(49,6)≈7.15×10−8\frac{1}{C(49,6)} \approx 7.15 \times 10^{-8}. What is the most likely explanation for the discrepancy?

  1. The sample size of 100,000 is insufficient to accurately estimate such a small probability
  2. The theoretical calculation is incorrect because it doesn't account for the order of selection
  3. The random number generator is not producing truly uniform distributions across the range 1-49
  4. The simulation contains a coding error that makes winning more likely than it should be (correct answer)
Explanation: When you encounter probability simulation problems, focus on comparing the observed results with theoretical expectations to identify systematic errors. The theoretical probability 1C(49,6)≈7.15×10−8\frac{1}{C(49,6)} \approx 7.15 \times 10^{-8} represents roughly 7 wins per 100 million trials. In 100,000 trials, you'd expect about 0.007 wins on average. However, the simulation shows 0.0000089 × 100,000 ≈ 0.89 wins, which is dramatically higher—about 125 times more frequent than expected. This massive discrepancy points to a systematic error making wins artificially likely, confirming answer D. Let's examine why the other options fail. Option A incorrectly assumes sample size is the issue, but even with sampling variation, getting 125 times the expected frequency is virtually impossible—this goes far beyond normal statistical fluctuation. Option B misunderstands combinatorics; since lottery numbers are drawn without replacement and order doesn't matter, C(49,6)C(49,6) correctly calculates the number of possible combinations. Option C suggests biased random number generation, but this would typically affect all outcomes similarly and wouldn't specifically increase the winning probability by such an extreme factor. The key insight is recognizing when observed results deviate so dramatically from theory that random variation cannot explain the difference. A coding error—such as incorrectly checking winning conditions, using the wrong comparison logic, or flawed random number generation specifically for the winning combination—could easily create this systematic bias toward false wins. Strategy tip: In simulation problems, small discrepancies suggest sampling error, but large discrepancies (especially when observed > theoretical) typically indicate systematic coding errors.

Question 2

A casino game simulation models a wheel with 38 equally likely outcomes (18 red, 18 black, 2 green). Players win on red or black but lose on green. The simulation runs 25,000 spins and estimates P(red or black) = 0.947. A second simulation of 100,000 spins estimates this probability as 0.946. Which statement best describes how to evaluate these simulation results?

  1. Both simulations overestimate because the theoretical probability 3638≈0.947\frac{36}{38} ≈ 0.947 should equal exactly 0.947
  2. The second simulation's estimate is definitively more accurate because 0.946 is closer to the theoretical value than 0.947
  3. The simulations are inconsistent and should be re-run because estimates from properly designed simulations should be identical
  4. The first simulation is less reliable despite similar estimates because random variation has greater impact with smaller samples (correct answer)
Explanation: When evaluating simulation results, you need to understand how sample size affects reliability and how random variation impacts estimates. The correct answer is D because sample size directly affects the reliability of simulation estimates. Even though both simulations produce similar results (0.947 and 0.946), the smaller sample of 25,000 spins is inherently less reliable due to greater random variation. With fewer trials, each individual outcome has more influence on the final estimate, making the result more susceptible to random fluctuations. The larger sample of 100,000 spins provides a more stable estimate because random variations tend to cancel out over more trials. Answer A is wrong because it misunderstands precision versus accuracy. The theoretical probability 3638≈0.9474\frac{36}{38} ≈ 0.9474, so 0.947 is actually quite close—simulations don't need to match theoretical values exactly due to random variation. Answer B incorrectly assumes that being closer to the theoretical value automatically means greater accuracy. While 0.946 happens to be closer to 0.9474, this doesn't make it definitively better—both estimates are well within expected variation ranges. Answer C misunderstands how simulations work. Different simulations will produce different estimates due to random variation—this is normal and expected, not a sign of poor design. Remember: when comparing simulations, larger sample sizes generally provide more reliable estimates, even if the actual numbers appear similar. The key insight is understanding that sample size affects reliability regardless of how close the estimates seem to each other.

Question 3

A genetics simulation models the probability that two parents, each carrying one recessive allele, will have at least one child with the recessive trait among their 3 children. Each child has a 25% chance of expressing the recessive trait. A student's simulation of 15,000 trials estimates this probability as 0.421. To improve the accuracy of this estimate, which modification would be most effective?

  1. Increase the number of trials to 45,000 while keeping all other parameters constant (correct answer)
  2. Change the recessive trait probability to 20% to account for incomplete penetrance
  3. Run multiple independent simulations of 15,000 trials each and average the results
  4. Modify the simulation to track 4 children instead of 3 to increase the sample size
Explanation: Increasing trials to 45,000 directly reduces the standard error of the estimate by a factor of √3, improving accuracy. The theoretical probability is 1-(0.75)³ ≈ 0.422, so the current estimate is already quite good. Choice B changes the problem being modeled. Choice C doesn't increase total trials. Choice D changes the research question rather than improving the estimate accuracy.

Question 4

An epidemiologist simulates disease spread in a population where each infected person has a 30% chance of infecting each of their 4 close contacts independently. She wants to estimate the probability that an infected person causes exactly 2 secondary infections. Her simulation uses 50,000 trials and produces an estimate with a 95% confidence interval of [0.264, 0.276]. How should she interpret this result?

  1. The simulation is highly accurate because the confidence interval is narrow, indicating low variability
  2. The result suggests each contact has a higher than 30% infection probability due to disease mutation
  3. The simulation needs more trials because the confidence interval doesn't include the theoretical value of 0.30
  4. The estimate should be compared to the theoretical binomial probability C(4,2)(0.3)2(0.7)2C(4,2)(0.3)^2(0.7)^2 to assess accuracy (correct answer)
Explanation: When you encounter simulation problems, focus on what's actually being measured and how to validate the results. This epidemiologist is simulating a binomial situation: each infected person has 4 independent contacts, each with a 30% infection probability. The correct approach is to compare the simulation results to the theoretical binomial probability. For exactly 2 secondary infections out of 4 contacts, this equals C(4,2)(0.3)2(0.7)2=6×0.09×0.49=0.2646C(4,2)(0.3)^2(0.7)^2 = 6 \times 0.09 \times 0.49 = 0.2646. Since this theoretical value (0.2646) falls within the confidence interval [0.264, 0.276], the simulation appears accurate. Answer D correctly identifies this validation process. Answer A misinterprets what the narrow confidence interval means. While it does indicate low variability due to the large sample size (50,000 trials), this alone doesn't prove accuracy—you must compare to the theoretical value. Answer B makes an unfounded leap about disease mutation. The confidence interval containing the theoretical probability actually suggests the model assumptions (30% infection rate) are working correctly, not that mutation is occurring. Answer C contains a critical error: the confidence interval DOES include the theoretical value of 0.2646 (not 0.30, which is the individual contact infection probability). This answer confuses the probability of exactly 2 infections with the probability per contact. Study tip: In simulation validation problems, always calculate the theoretical probability first, then check if it falls within the confidence interval. Don't confuse individual event probabilities with compound event probabilities—they measure different things entirely.

Question 5

A game involves rolling two dice and winning if the sum is 7 or 11. Maria runs a simulation with 1,000 trials and estimates the winning probability as 0.228. Her friend Carlos runs the same simulation with 5,000 trials and estimates it as 0.222. The theoretical probability is 836=0.222‾\frac{8}{36} = 0.\overline{222}. Which statement best evaluates the accuracy of these simulations?

  1. Maria's estimate is more accurate because it's closer to the theoretical value in absolute terms
  2. Carlos's estimate is more accurate because larger sample sizes generally produce more reliable estimates (correct answer)
  3. Both estimates are equally accurate since they're within 0.01 of the theoretical probability
  4. Neither estimate can be considered accurate without knowing the standard deviation of the sampling distribution
Explanation: Larger sample sizes generally provide more reliable estimates due to the Law of Large Numbers, even when a smaller sample happens to be closer to the true value by chance. Carlos's 5,000 trials provide a more reliable basis for estimation than Maria's 1,000 trials. Choice A focuses only on proximity, ignoring sample size reliability. Choice C arbitrarily defines accuracy. Choice D overcomplicates the basic principle of sample size and reliability.

Question 6

A meteorologist simulates the probability of rain on at least 3 days during a 5-day period, where each day has a 60% chance of rain independently. Two different simulation approaches are tested: Method A generates 5 random numbers per trial, Method B uses a binomial random number generator. After 20,000 trials each, Method A estimates 0.663 while Method B estimates 0.682. What should the meteorologist conclude?

  1. Method A is more accurate because 0.663 is closer to the theoretical probability of 0.65
  2. Method B is preferable because binomial generators are always more accurate than sequential random number generation
  3. Both methods should produce similar estimates; the difference suggests a possible implementation error to investigate (correct answer)
  4. The difference is expected because Method A simulates weather patterns while Method B uses pure probability theory
Explanation: Both methods should estimate the same theoretical probability (1 - P(0,1,or 2 rainy days) using binomial distribution). With 20,000 trials, such different estimates (0.663 vs 0.682) suggest implementation errors in one or both methods. Choice A incorrectly assumes 0.65 is the theoretical value. Choice B overgeneralizes about method superiority. Choice D incorrectly distinguishes between the methods' theoretical basis.

Question 7

A computer security analyst simulates network attacks where an attacker needs to succeed at 2 consecutive attempts, each with a 40% success rate, before being locked out after any failure. The simulation runs 30,000 trials and estimates the probability of a successful breach as 0.159. Which evaluation of this simulation result is most appropriate?

  1. The result is accurate because 0.159 ≈ (0.40)² = 0.16 for two consecutive successes
  2. The result underestimates the probability because it should equal 0.40 + 0.40 = 0.80 for two attempts
  3. The result is reasonable and close to the theoretical probability of successful breach under these conditions (correct answer)
  4. The result overestimates because it doesn't properly account for the lockout mechanism after failures
Explanation: The theoretical probability is indeed (0.40)² = 0.16 for two consecutive successes, and 0.159 is very close to this value, indicating a reasonable simulation result. Choice A makes the right calculation but doesn't fully evaluate the simulation quality. Choice B incorrectly adds probabilities. Choice D misunderstands that the lockout after failure is already built into requiring consecutive successes.

Question 8

A quality control manager wants to estimate the probability that exactly 2 out of 5 randomly selected items from a production line are defective. She designs a simulation where each trial involves generating 5 random numbers between 0 and 1, with values less than 0.15 representing defective items. After running 10,000 trials, she finds that 1,382 trials resulted in exactly 2 defective items. What is the most accurate interpretation of this simulation result?

  1. The actual defect rate in the production line is approximately 13.82%
  2. The probability of exactly 2 defective items in a sample of 5 is approximately 0.1382 (correct answer)
  3. The simulation is flawed because 1,382 is too close to the expected theoretical value
  4. The manager should increase the number of trials to exactly 15,000 for better accuracy
Explanation: The simulation estimates P(exactly 2 defective in 5 items) = 1,382/10,000 = 0.1382. Choice A confuses the defect rate (15%) with the compound probability being estimated. Choice C incorrectly suggests proximity to theoretical values indicates a flaw. Choice D arbitrarily suggests a specific sample size without justification.

Question 9

A biologist wants to simulate the probability that a rare bird species will successfully nest in a particular habitat over a 3-year period. Historical data suggests a 40% annual nesting success rate, with each year's outcome independent of previous years. She uses a simulation where random numbers 0.00-0.39 represent successful nesting. After 2,500 trials, she observes 768 trials where the bird successfully nested in all 3 years. What does this suggest about her simulation design?

  1. The simulation correctly estimates P(success in all 3 years) ≈ 0.307, which matches the theoretical value
  2. The simulation overestimates the probability because 768/2,500 = 0.307 > 0.40³ = 0.064 (correct answer)
  3. The simulation is flawed because it should use integers rather than decimal ranges for random number generation
  4. The simulation underestimates the probability because it doesn't account for the habitat improving over time
Explanation: The theoretical probability of success in all 3 independent years is 0.40³ = 0.064. The simulation result gives 768/2,500 = 0.307, which is much higher than expected, suggesting an error in implementation. Choice A incorrectly states the theoretical value. Choice C incorrectly criticizes the use of decimals. Choice D introduces irrelevant factors not mentioned in the problem setup.

Question 10

A quality assurance engineer simulates a process where components are tested until the first defective one is found, with each component having a 12% defect rate. She wants to estimate the probability that the first defective component is found on the 4th test. Her simulation of 60,000 trials gives an estimate of 0.085. Which statement best describes how she should validate this result?

  1. Compare the estimate to (0.88)3×0.12≈0.082(0.88)^3 \times 0.12 ≈ 0.082 and assess whether 0.085 represents reasonable sampling variation (correct answer)
  2. Verify that exactly 12% of all simulated components were classified as defective across all trials
  3. Check that the average position of the first defective component across trials is approximately 10.12≈8.33\frac{1}{0.12} ≈ 8.33
  4. Confirm that the simulation stopped after finding exactly one defective component in each of the 60,000 trials
Explanation: The theoretical probability for a geometric distribution is (0.88)³ × 0.12 ≈ 0.082 (3 successes then 1 failure). Her estimate of 0.085 is very close, representing reasonable sampling variation. Choice B checks overall defect rate but not the specific probability being estimated. Choice C validates a different parameter (expected value). Choice D confirms simulation logic but doesn't validate the probability estimate.

Question 11

A researcher designs a simulation to estimate the probability that at least one of four independent medical tests will give a false positive result. Each test has a 5% false positive rate. She runs 8,000 trials and wants to achieve a margin of error of approximately ±0.01 with 95% confidence. Her simulation yields an estimate of 0.186. Which aspect of her simulation setup should she evaluate first?

  1. Whether 8,000 trials provides sufficient precision for the desired margin of error of ±0.01
  2. Whether the estimate 0.186 is reasonable compared to the theoretical probability of approximately 0.185 (correct answer)
  3. Whether the random number generator properly simulates the 5% false positive rate for each test
  4. Whether the independence assumption between tests is correctly implemented in the simulation code
Explanation: First, she should verify if 0.186 is reasonable. The theoretical probability is 1 - (0.95)⁴ ≈ 0.185, so her estimate appears very close. Choice A about sample size is secondary if the estimate seems wrong. Choices C and D are implementation details to check only after confirming the estimate makes sense theoretically.

Question 12

A pharmaceutical researcher simulates clinical trial outcomes where a drug has a 75% success rate. She models trials with 12 patients each and wants to estimate the probability that exactly 8 patients show improvement. Her simulation of 40,000 trials yields an estimate of 0.168. She calculates the theoretical probability as C(12,8)(0.75)8(0.25)4≈0.195C(12,8)(0.75)^8(0.25)^4 ≈ 0.195. What should she investigate first?

  1. Whether the random number generator is producing the correct 75% success rate across all simulated patients (correct answer)
  2. Whether 40,000 trials is sufficient for estimating probabilities in the 0.15-0.20 range with adequate precision
  3. Whether the theoretical calculation correctly accounts for the discrete nature of patient outcomes
  4. Whether the simulation properly implements independent outcomes for each of the 12 patients within each trial
Explanation: The simulation estimate (0.168) is notably lower than the theoretical value (0.195), suggesting the random number generator might not be producing the intended 75% success rate. This is the most fundamental check. Choice B addresses precision but not the systematic difference. Choice C questions the correct theoretical model. Choice D is important but secondary to verifying the basic probability implementation.

Question 13

A basketball player claims to make 70% of her free throws. To test this claim, a coach designs a simulation using a random number generator that produces integers from 1 to 10. The simulation runs 500 trials, where each trial represents one free throw attempt. If the simulation estimates the player makes 68% of her free throws, which assignment of numbers to outcomes would be most appropriate for this simulation?

  1. Numbers 1-7 represent made shots, 8-10 represent missed shots (correct answer)
  2. Numbers 1-6 represent made shots, 7-10 represent missed shots
  3. Numbers 1-68 represent made shots, 69-100 represent missed shots
  4. Numbers 1-70 represent made shots, 71-100 represent missed shots
Explanation: For a simulation to test a 70% success rate, we need the probability assignment to match 70%. With integers 1-10, numbers 1-7 give us 7/10 = 0.70 = 70%. Choice B gives 60%, Choice C would require numbers 1-100, and Choice D also requires numbers 1-100. The simulation result of 68% is the outcome, not the setup parameter.

Question 14

A sports analyst simulates basketball games where Team A has a 65% chance of winning each game. She runs simulations to estimate the probability that Team A wins exactly 4 out of 7 games in a playoff series. After 50,000 trials, her simulation estimates this probability as 0.244. She then runs an additional 50,000 trials and gets an estimate of 0.239. How should she use these results to assess the accuracy of her simulation?

  1. Average the two estimates to get 0.2415, which provides a more accurate result than either individual simulation
  2. Use only the second estimate (0.239) because it benefits from her experience improving the simulation code
  3. Compare both estimates to the theoretical binomial probability C(7,4)(0.65)4(0.35)3C(7,4)(0.65)^4(0.35)^3 to evaluate consistency (correct answer)
  4. Conclude that the simulation is unreliable because two runs produced different estimates (0.244 vs 0.239)
Explanation: She should compare both estimates to the theoretical value C(7,4)(0.65)⁴(0.35)³ ≈ 0.244 to assess accuracy. Both estimates are very close to this theoretical value and to each other, indicating good simulation performance. Choice A is reasonable but doesn't assess accuracy. Choice B assumes unjustified improvement. Choice D misunderstands that different estimates from random sampling are normal and expected.

Question 15

A factory quality inspector simulates the number of defective items in batches of 50, where each item has a 4% defect rate. She runs 500 simulations and wants to estimate the probability that a batch contains more than 3 defective items. The results show the following distribution: 0 defects (72 batches), 1 defect (144 batches), 2 defects (138 batches), 3 defects (89 batches), 4 defects (37 batches), 5+ defects (20 batches). What factors most significantly limit the accuracy of this probability estimate?

  1. The sample size of 500 simulations is too small for reliable estimation, and the 4% defect rate creates computational rounding errors in the simulation
  2. The simulation design fails to account for potential correlation between defective items, and the batch size of 50 is insufficient for accurate modeling
  3. The categorization of '5+ defects' loses important information, and the assumed independence of defects may not reflect real manufacturing processes
  4. The primary limitation is the moderate sample size of 500 trials, while the simulation design appropriately models independent defects at the specified rate (correct answer)
Explanation: When evaluating simulation accuracy, you need to consider both sample size adequacy and whether the modeling assumptions match reality. This question tests your ability to identify the primary factors that limit estimation precision in a binomial simulation. The correct answer is D because 500 simulations provides a reasonably robust sample size for estimating tail probabilities. While larger samples always improve precision, 500 trials is sufficient to detect meaningful patterns, especially when estimating probabilities in the 10-15% range (more than 3 defects). The simulation design appropriately models the scenario as independent Bernoulli trials with a fixed 4% defect rate, which is the standard approach for quality control modeling. Answer A incorrectly suggests 500 simulations is inadequate and that 4% creates rounding errors. Modern computers handle 4% probability calculations without significant rounding issues, and 500 trials is reasonable for this estimation. Answer B wrongly claims the batch size of 50 is insufficient. Batch size of 50 with 4% defect rate creates a proper binomial distribution, and there's no indication that defect correlation is a primary concern in this simulation context. Answer C overstates the impact of grouping "5+ defects" together. Since this category represents only 20 out of 500 simulations (4%), the lost granularity has minimal effect on estimating P(X > 3). While defect independence might not perfectly reflect reality, this is a secondary consideration compared to sample size. Study tip: In simulation problems, focus first on sample size adequacy, then examine whether the modeling assumptions reasonably approximate the real scenario. Don't get distracted by theoretical perfection when practical adequacy is the standard.

Question 16

A meteorologist simulates weather patterns to estimate the probability that it rains on exactly 3 out of 7 consecutive days, given a daily rain probability of 40%. Two different simulation approaches are proposed: Method A uses random integers 1-10 where 1-4 represent rain, and Method B uses random decimals 0-1 where values ≤ 0.4 represent rain. After 1000 trials, Method A gives 290 successes and Method B gives 307 successes. What is the most accurate assessment of these results?

  1. Method B is superior because decimal random numbers provide inherently better precision than integer-based random selection methods
  2. Both methods are mathematically equivalent and properly model 40% probability, so the difference in results reflects normal sampling variation (correct answer)
  3. Method A is more reliable because integer-based simulations eliminate rounding errors that can accumulate in decimal-based probability calculations
  4. The results suggest Method B is more accurate since 307/1000 = 0.307 is closer to the theoretical probability than 290/1000 = 0.290
Explanation: Both simulation methods correctly model a 40% rain probability. Method A: 4 out of 10 integers = 40%. Method B: decimals ≤ 0.4 = 40%. The difference between 290 and 307 successes (1.7 percentage points) is well within normal sampling variation for 1000 trials. Neither method is inherently superior in precision, and being closer to any particular theoretical value in one run doesn't indicate greater accuracy.

Question 17

A researcher designs a simulation to estimate the probability that in a group of 25 people, at least 3 share the same birthday. The simulation uses random integers from 1 to 365 to represent birthdays, ignoring leap years. After 2000 trials, 1340 trials showed at least 3 people sharing a birthday. A colleague suggests the simulation would be more accurate if it used 1000 trials instead of 2000. How should the researcher respond?

  1. The colleague is correct because smaller sample sizes reduce computational errors and provide more reliable estimates in birthday probability problems
  2. The colleague is incorrect because larger sample sizes generally provide more accurate probability estimates, assuming the simulation design is sound (correct answer)
  3. The colleague's suggestion has merit since 1000 trials would eliminate potential bias from the current oversampling of rare birthday combinations
  4. The sample size change would be beneficial because it better matches the theoretical framework for birthday problems involving multiple shared dates
Explanation: The colleague is incorrect. In Monte Carlo simulations, larger sample sizes generally provide more accurate estimates by reducing sampling error, assuming the simulation is properly designed. The current simulation appropriately models the scenario using random integers 1-365 for birthdays. Using 2000 trials instead of 1000 gives a more reliable estimate of 1340/2000 = 0.67. There's no computational error issue or bias from 'oversampling' with larger sample sizes.

Question 18

Two students simulate the probability that exactly 4 heads occur in 10 coin flips. Student A uses a random number generator producing values 0.0 to 1.0, where ≤0.5 represents heads. Student B uses the same generator but assigns ≤0.49 as heads to 'account for slight bias in real coins.' After 1200 trials each, Student A reports 251 successes and Student B reports 201 successes. Which analysis is most appropriate?

  1. Student A's simulation is properly calibrated for the theoretical problem, while Student B's modification introduces unnecessary bias that skews results downward (correct answer)
  2. Student B's approach better models realistic conditions, and the lower success count of 201 suggests this adjustment reveals the true probability more accurately
  3. Both simulations are valid modeling approaches, but Student A's higher success rate indicates a systematic error in the random number generation process
  4. Student B's method is more sophisticated because it accounts for real-world variability, making the estimate of 201/1200 ≈ 0.168 more reliable than Student A's 0.209
Explanation: When evaluating simulation design, you need to distinguish between modeling the theoretical problem versus introducing modifications that change the underlying probability structure. Student A correctly models the theoretical scenario of fair coin flips by using p = 0.5 for heads. The probability of exactly 4 heads in 10 flips follows a binomial distribution: P(X=4)=(104)(0.5)10≈0.205P(X=4) = \binom{10}{4}(0.5)^{10} \approx 0.205. Student A's result of 251/1200 ≈ 0.209 aligns well with this theoretical value, indicating proper calibration. Student B's modification to p = 0.49 fundamentally changes the problem being simulated. While real coins might have slight bias, the question asks about simulating the probability for the theoretical case of exactly 4 heads in 10 flips. By reducing the probability of heads, Student B creates a different scenario entirely, making the simulation less likely to produce exactly 4 heads. Choice A correctly identifies that Student A properly models the theoretical problem while Student B introduces unnecessary bias. Choice B incorrectly suggests Student B's approach is more realistic when the question requires modeling the theoretical scenario. Choice C wrongly attributes Student A's higher success rate to systematic error rather than correct calibration. Choice D mistakenly calls Student B's method "sophisticated" when it actually misaligns with the problem requirements. Study tip: In simulation problems, always identify whether you're modeling a theoretical scenario (use exact probabilities) or a real-world situation (where modifications might be justified). Don't add complexity unless the problem specifically calls for it.

Question 19

A student simulates rolling two dice 100 times to estimate the probability of getting a sum greater than 9. She uses a random number table to generate pairs of single digits, ignoring any digits 0, 7, 8, or 9. After filtering, she obtains 60 valid pairs and finds that 12 pairs sum to more than 9. She estimates the probability as 12/60 = 0.2. Which statement best describes the quality of this simulation?

  1. The simulation design is flawed because filtering out 0, 7, 8, 9 creates unequal probabilities for the remaining digits, biasing the results toward lower sums
  2. The simulation is well-designed and the estimate of 0.2 is accurate, but precision would improve significantly with a larger initial sample size
  3. The filtering method is appropriate and maintains equal probabilities, but 60 trials provides insufficient data for a reliable probability estimate (correct answer)
  4. The simulation correctly models dice rolls through appropriate filtering, yielding a reasonable estimate that matches theoretical expectations for this scenario
Explanation: The filtering method is appropriate: using digits 1-6 correctly models standard dice with equal probabilities. The theoretical probability of sum > 9 is 6/36 = 1/6 ≈ 0.167, so 0.2 is reasonable given sampling variation. However, 60 trials is a relatively small sample size for probability estimation, limiting reliability. The filtering doesn't bias toward lower sums since it maintains uniform distribution among 1-6.

Question 20

A medical researcher simulates treatment outcomes where each patient has a 70% recovery rate. She wants to estimate the probability that in a group of 15 patients, between 8 and 12 patients (inclusive) recover. Using random decimals, she runs 800 trials where decimals ≤0.70 represent recovery. The simulation yields 428 trials meeting the criteria. She plans to report this as strong evidence that the probability is approximately 0.535. What is the most significant concern with this conclusion?

  1. The simulation design is flawed because using ≤0.70 creates a 71% recovery rate rather than the intended 70%, systematically biasing results upward
  2. The sample size of 800 trials provides insufficient precision for medical research applications, requiring at least 5000 trials for reliable probability estimates
  3. The estimate of 0.535 lacks context about uncertainty, and 800 trials provides reasonable but limited precision that should be acknowledged in medical reporting (correct answer)
  4. The conclusion incorrectly treats a single simulation result as definitive evidence rather than recognizing it as one estimate subject to sampling variation
Explanation: The main concern is that the researcher plans to report 0.535 as 'strong evidence' without acknowledging the uncertainty inherent in simulation estimates. While 800 trials provides a reasonable estimate, it has limited precision and the result should be presented with appropriate caveats about sampling variation. The simulation design is actually correct (decimals ≤0.70 properly represents 70%). Choice D overstates the issue - simulation estimates can be valuable evidence when properly contextualized.