College Statistics Quiz: Simulation For Probability
20 questions · exam conditions
0:00
Simulation For ProbabilityQuestion 1 of 20

A city council has 9 members: 5 from Party X and 4 from Party Y. A committee of 3 is to be selected at random. A student wants to simulate the probability that the committee is composed entirely of members from Party X. Which simulation procedure using a standard 52-card deck is valid for one trial?

Take 5 red cards (Party X) and 4 black cards (Party Y). Shuffle this packet of 9 cards, draw one card, and note its color. Return the card, reshuffle, and repeat two more times. If all three cards are red, record a success.
Take 5 red cards (Party X) and 4 black cards (Party Y). Shuffle this packet of 9 cards and deal three cards from the top. If all three cards are red, record a success.
Designate Aces as Party Y (4 cards) and Kings as Party X (4 cards). Shuffle the full deck and draw 3 cards. If all 3 cards are Kings, record a success for the trial.
Draw 9 cards from the full deck. If there are exactly 5 red and 4 black cards, keep them. Otherwise, reshuffle the full deck and repeat until the condition is met. Then, draw 3 from the 9 cards. If all are red, record a success.
← Back to quizzes

College Statistics Quiz

College Statistics Quiz: Simulation For Probability

Practice Simulation For Probability in College Statistics 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 Simulation For Probability, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

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 city council has 9 members: 5 from Party X and 4 from Party Y. A committee of 3 is to be selected at random. A student wants to simulate the probability that the committee is composed entirely of members from Party X. Which simulation procedure using a standard 52-card deck is valid for one trial?

  1. Take 5 red cards (Party X) and 4 black cards (Party Y). Shuffle this packet of 9 cards, draw one card, and note its color. Return the card, reshuffle, and repeat two more times. If all three cards are red, record a success.
  2. Take 5 red cards (Party X) and 4 black cards (Party Y). Shuffle this packet of 9 cards and deal three cards from the top. If all three cards are red, record a success. (correct answer)
  3. Designate Aces as Party Y (4 cards) and Kings as Party X (4 cards). Shuffle the full deck and draw 3 cards. If all 3 cards are Kings, record a success for the trial.
  4. Draw 9 cards from the full deck. If there are exactly 5 red and 4 black cards, keep them. Otherwise, reshuffle the full deck and repeat until the condition is met. Then, draw 3 from the 9 cards. If all are red, record a success.
Explanation: Forming a committee is a selection process without replacement. The population consists of 9 individuals (5 from X, 4 from Y). Option B correctly models this by creating a representative deck of 9 cards (5 red for X, 4 black for Y). Shuffling and dealing three cards without replacement perfectly mimics the random selection of the 3 committee members. A success is correctly defined as all three drawn cards being red.

Question 2

A company has two production lines, A and B. Line A produces 60% of phones and has a 5% defect rate. Line B produces 40% of phones and has a 10% defect rate. A simulation is designed to estimate the probability that a randomly selected defective phone came from Line A. Which simulation plan would correctly estimate this probability?

  1. Simulate a large number of phones. For each, assign it to Line A with probability 0.6. Then, determine if it is defective with probability 0.05. The result is the proportion of all simulated phones that are defective and from Line A.
  2. Simulate selecting a defective phone by first choosing a line (60% A, 40% B) and then checking a second condition for the defect rate. Repeat many times and find the proportion of trials where the line chosen was A.
  3. Simulate a large number of phones according to their production and defect rates. Then, considering only the phones that were marked as defective, calculate the proportion that came from Line A. (correct answer)
  4. Simulate an equal number of phones from Line A and Line B. For each group, use the respective defect rates to find the number of defects. The result is (defects from A) / (total defects from A + B).
Explanation: The question asks for a conditional probability: P(from Line A | phone is defective). A direct simulation of this requires creating a large population of phones that reflects the overall production process. After simulating many phones, we must first isolate the sub-population of interest—all the defective phones. The estimated probability is then the number of defective phones from Line A divided by the total number of defective phones from both lines.

Question 3

A company's call center has two representatives, Alice and Bob. Alice successfully resolves 90% of her calls, while Bob resolves 70%. Alice takes 60% of all calls. A simulation is planned to estimate the overall resolution rate. The plan is:

  1. Generate a random digit 1-10. If 1-6, the call goes to Alice. If 7-10, to Bob.

  2. If Alice, generate a second digit. If 1-9, it's resolved.

  3. If Bob, generate a second digit. If 1-7, it's resolved. This simulation model implicitly assumes which of the following?

  1. The number of calls Alice and Bob take in a day is approximately equal.
  2. A call's difficulty is independent of which representative receives it. (correct answer)
  3. The overall resolution rate is simply the average of Alice's and Bob's individual rates.
  4. The probability of a call being resolved is independent of whether Alice or Bob takes it.
Explanation: The simulation first assigns a call to a representative and then determines the outcome based on that representative's fixed success rate. This structure assumes that the initial assignment of the call (and thus its underlying difficulty) is random and does not depend on the representative. If, for example, more difficult calls were systematically routed to Alice, her true success rate for an average call might be different, and this model would be invalid. The model assumes the pool of calls each representative draws from is probabilistically identical.

Question 4

At a holiday gift exchange with 6 participants, each person brings one gift. The gifts are placed in a pile, mixed, and then each person randomly draws one gift. A simulation is designed to estimate the probability that at least one person draws their own gift. Which of the following procedures correctly models one trial of this simulation?

  1. Roll a 6-sided die 6 times. Let the sequence represent the gifts drawn by persons 1 through 6. Check if any number matches its position in the sequence (e.g., a '1' on the first roll).
  2. Create a list of numbers 1, 2, 3, 4, 5, 6. Randomly select a number from the list 6 times, with replacement, and record if any number is selected more than once.
  3. Take 6 index cards and number them 1 through 6. Shuffle the cards and arrange them in a line. Count the number of cards for which the card's number matches its position in the line (e.g., card '3' is in the 3rd position). (correct answer)
  4. Roll a 6-sided die. If it's a 1, count it as a success for one person drawing their own gift. Repeat 6 times and divide the number of successes by 6.
Explanation: This problem is a classic derangement or matching problem. Each person must draw one unique gift, meaning the process is without replacement. Procedure C correctly models this. Numbering the people and their corresponding gifts from 1 to 6, shuffling the numbered cards is equivalent to randomly distributing the gifts. Checking if a card's number matches its position is equivalent to checking if a person drew their own gift. This entire process constitutes one valid trial.

Question 5

A fast-food chain offers one of four different toys with a kids' meal. The probabilities of getting each toy are not equal: Toy A (40%), Toy B (30%), Toy C (20%), and Toy D (10%). A researcher wants to simulate the number of meals one must purchase to collect all four toys. Using a random number table with digits 0-9, which of the following describes a valid assignment of digits and procedure for one trial?

  1. Assign 1 to Toy A, 2 to Toy B, 3 to Toy C, and 4 to Toy D, ignoring digits 0 and 5-9. Generate digits from the valid set until 1, 2, 3, and 4 have all been selected. The number of digits generated is the outcome of the trial.
  2. Assign 0-3 to Toy A, 4-6 to Toy B, 7-8 to Toy C, and 9 to Toy D. Generate digits until at least one digit from each of the four assigned groups has been selected. The number of digits generated represents the number of meals purchased. (correct answer)
  3. Assign 0-3 to Toy A, 4-6 to Toy B, 7-8 to Toy C, and 9 to Toy D. Generate a set of 10 random digits. The number of distinct toy groups represented in the set of 10 digits is the outcome of the trial.
  4. Assign 0-4 to Toy A, 5-7 to Toy B, 8 to Toy C, and 9 to Toy D. Generate digits until a digit is repeated. The number of unique digits generated before the repeat represents the number of unique toys collected.
Explanation: A valid simulation must correctly model the probabilities of the events. The assignment in B (0-3 for Toy A, 4-6 for Toy B, 7-8 for Toy C, 9 for Toy D) correctly represents the 40%, 30%, 20%, and 10% probabilities. A trial consists of simulating purchases until all four toys are collected, which means generating digits until one from each category has appeared. The number of digits generated correctly corresponds to the number of meals.

Question 6

A manufacturing process has a 2% defect rate. The company ships products in boxes of 50. A simulation is designed to estimate the probability that a box contains two or more defective items. The simulation uses a random number generator that produces numbers between 0 and 1. For a single item, a number less than 0.02 represents a defect. How should one trial of the simulation be structured?

  1. Generate random numbers until two numbers less than 0.02 are generated. The number of random numbers generated is the outcome of the trial.
  2. Generate 50 random numbers. If the sum of the numbers is less than 1.0 (50 * 0.02), record a success for the trial.
  3. Generate 50 random numbers, one for each item in the box. Count how many of these numbers are less than 0.02. If this count is two or more, record a success for the trial. (correct answer)
  4. Generate a single random number. If it is between 0.02 and 0.04, record a success for the trial, as this range represents the probability of exactly two defects.
Explanation: A single trial must simulate the inspection of one entire box of 50 items. Option C correctly models this. It generates 50 random numbers, representing the 50 items. It uses the correct rule (number < 0.02) to identify a defect for each item. Finally, it correctly counts the total number of defects in the simulated box and applies the success condition for the trial (count ≥ 2).

Question 7

A researcher is simulating a family having children until they have one boy, but they will stop after a maximum of 4 children regardless of the outcome. The probability of having a boy is 0.5. The simulation is run 1000 times. The results for the number of children in the family are: 1 child (495 times), 2 children (254 times), 3 children (128 times), 4 children (123 times). What is the simulated probability that a family in this scenario has a boy?

  1. 0.500
  2. 0.877 (correct answer)
  3. 1.000
  4. 0.938
Explanation: Based on the simulation design, families stop when they have a boy OR when they reach 4 children. Families with 1, 2, or 3 children stopped because they had a boy (that's the only way to stop before the 4-child limit). This accounts for 495 + 254 + 128 = 877 families that definitely have a boy. The 123 families with 4 children reached the maximum limit, which means their first 3 children were girls. The simulation results don't specify the gender of the 4th child for these families, so we cannot count them as having boys based solely on the given data. Therefore, the simulated probability that a family has a boy is 877/1000 = 0.877.

Question 8

A game involves rolling a pair of standard six-sided dice. A player wins on the first roll if the sum is 7. They lose if the sum is 2, 3, or 12. If they roll any other sum, they roll again until they either win (by rolling a 7) or lose (by rolling a 2, 3, or 12). A simulation is set up to estimate the probability of winning. A trial consists of generating pairs of random integers (1-6) until the game ends. Using the sequence of simulated dice rolls below, what are the outcomes of the first three trials?

Sequence of rolls: (3, 4), (5, 6), (1, 1), (6, 2), (4, 4), (5, 2), (6, 6)

  1. Win, Lose, Win (correct answer)
  2. Win, Re-roll, Lose
  3. Win, Lose, Re-roll
  4. Win, Lose, Lose
Explanation: We must trace the game for each trial. A trial ends only with a Win (sum 7) or a Loss (sum 2, 3, or 12).
  • Trial 1: The first roll is (3, 4), sum = 7. This is a Win. The trial ends.
  • Trial 2: The next roll is (5, 6), sum = 11. This is a re-roll. The next roll is (1, 1), sum = 2. This is a Loss. The trial ends.
  • Trial 3: The next roll is (6, 2), sum = 8. This is a re-roll. The next roll is (4, 4), sum = 8. This is another re-roll. The next roll is (5, 2), sum = 7. This is a Win. The trial ends. The outcomes of the first three trials are Win, Lose, Win.

Question 9

Four researchers are simulating the probability of getting heads on a weighted coin, for which the true probability is 0.6. Each researcher runs a different number of trials. Researcher A runs 20 trials, B runs 100, C runs 500, and D runs 2000. Which researcher's simulation is most likely to produce an estimated probability that is farthest from the true value of 0.6?

  1. Researcher A (correct answer)
  2. Researcher B
  3. Researcher C
  4. Researcher D
Explanation: According to the Law of Large Numbers, as the number of trials in a simulation increases, the estimated probability tends to get closer to the true theoretical probability. Conversely, simulations with fewer trials have higher sampling variability, meaning their results are more spread out and more likely to be far from the true value. Researcher A, with only 20 trials, has the smallest sample size and is therefore most likely to obtain a result that deviates significantly from 0.6.

Question 10

A botanist knows that the probability of a certain plant producing a flower with a desirable trait is 0.20. She wants to simulate the number of plants she must inspect to find the first one with the trait. Using a random number generator that produces integers from 1 to 10, which of the following describes a valid setup for a single trial?

  1. Let integers 1 and 2 represent success and 3-10 represent failure. Generate random integers until a 1 or 2 appears. The total number of integers generated is the result for the trial. (correct answer)
  2. Let integers 1 and 2 represent success and 3-10 represent failure. Generate a sample of 10 random integers. Count how many of the integers are a 1 or a 2.
  3. Let the integer 2 represent success and all other integers represent failure. Generate integers until a 2 appears. The total number of integers generated is the result for the trial.
  4. Let integer 1 represent success and 2-5 represent failure. Ignore integers 6-10. Generate random integers from the valid set, and the trial ends when a 1 appears.
Explanation: The simulation needs to model a geometric setting: waiting for the first success. The probability of success is 0.20. Option A correctly assigns 2 out of 10 integers (1 and 2) to success, making P(Success) = 2/10 = 0.20. A trial correctly consists of generating numbers until the first success occurs, and the outcome is the number of trials needed. This procedure accurately models the scenario and efficiently uses all generated digits.

Question 11

A circular dartboard has a radius R. In its center is a smaller circular bullseye with a radius of R/4. Assuming a dart hits a random point on the dartboard, a simulation is run to estimate the probability of hitting the bullseye. The simulation generates 2000 random points (x, y) in a square of side 2R that encloses the dartboard. It counts a 'hit' if the point is within the dartboard (x² + y² ≤ R²) and a 'bullseye' if the point is within the bullseye (x² + y² ≤ (R/4)²). The simulation produced 1572 hits on the dartboard, of which 96 were in the bullseye. What is the estimated probability of hitting the bullseye, given the dart hit the dartboard?

  1. 0.048
  2. 0.0625
  3. 0.786
  4. 0.061 (correct answer)
Explanation: The question asks for an estimated probability from the simulation, conditional on the dart hitting the dartboard. This means our sample space is not the 2000 points generated, but only the 1572 points that successfully landed on the dartboard. Within this sample space, 96 points also landed in the bullseye. Therefore, the estimated conditional probability is the ratio of the number of bullseyes to the number of dartboard hits: 96 / 1572 ≈ 0.061.

Question 12

In a carnival game, a player tosses a ring at a board of 20 pegs. The player has a 30% chance of hooking a peg with any given toss. To estimate the probability of needing exactly 4 tosses to get the first success, a simulation is proposed. Which of the following correctly describes how to use a random number table to find the outcome of a single trial?

  1. Use two-digit numbers 00-99. Let 00-29 be a success. Generate 4 numbers. If exactly one of them is in the success range, the trial is a success.
  2. Use two-digit numbers 00-99. Let 00-29 be a success. Generate numbers one at a time. The trial's outcome is a 'yes' if the first success occurs on the fourth number generated.
  3. Use single digits 0-9. Let 0-2 be a success. Generate a sequence of digits. The number of digits generated until the first success is the result of the trial. Repeat many times and count how often the result is 4. (correct answer)
  4. Use single digits 0-9. Let 0-2 be a success. Generate 4 digits. The result of the trial is the number of successes found within those 4 digits.
Explanation: This is a geometric probability scenario: waiting for the first success. A trial should simulate this waiting process. Option C correctly sets up the probability of success (3 out of 10 digits = 30%) and defines a trial as generating digits until the first success occurs. The value recorded for the trial is the number of tosses needed. To find the desired probability, one would run many such trials and then calculate the proportion of trials where the recorded value was exactly 4.

Question 13

A fast-food chain offers one of four different toys with a kids' meal. The probabilities of getting each toy are not equal: Toy A (40%), Toy B (30%), Toy C (20%), and Toy D (10%). A researcher wants to simulate the number of meals one must purchase to collect all four toys. Using a random number table with digits 0-9, which of the following describes a valid assignment of digits and procedure for one trial?

  1. Assign 1 to Toy A, 2 to Toy B, 3 to Toy C, and 4 to Toy D, ignoring digits 0 and 5-9. Generate digits from the valid set until 1, 2, 3, and 4 have all been selected. The number of digits generated is the outcome of the trial.
  2. Assign 0-3 to Toy A, 4-6 to Toy B, 7-8 to Toy C, and 9 to Toy D. Generate digits until at least one digit from each of the four assigned groups has been selected. The number of digits generated represents the number of meals purchased. (correct answer)
  3. Assign 0-3 to Toy A, 4-6 to Toy B, 7-8 to Toy C, and 9 to Toy D. Generate a set of 10 random digits. The number of distinct toy groups represented in the set of 10 digits is the outcome of the trial.
  4. Assign 0-4 to Toy A, 5-7 to Toy B, 8 to Toy C, and 9 to Toy D. Generate digits until a digit is repeated. The number of unique digits generated before the repeat represents the number of unique toys collected.
Explanation: A valid simulation must correctly model the probabilities of the events. The assignment in B (0-3 for Toy A, 4-6 for Toy B, 7-8 for Toy C, 9 for Toy D) correctly represents the 40%, 30%, 20%, and 10% probabilities. A trial consists of simulating purchases until all four toys are collected, which means generating digits until one from each category has appeared. The number of digits generated correctly corresponds to the number of meals.

Question 14

Four researchers are simulating the probability of getting heads on a weighted coin, for which the true probability is 0.6. Each researcher runs a different number of trials. Researcher A runs 20 trials, B runs 100, C runs 500, and D runs 2000. Which researcher's simulation is most likely to produce an estimated probability that is farthest from the true value of 0.6?

  1. Researcher A (correct answer)
  2. Researcher B
  3. Researcher C
  4. Researcher D
Explanation: According to the Law of Large Numbers, as the number of trials in a simulation increases, the estimated probability tends to get closer to the true theoretical probability. Conversely, simulations with fewer trials have higher sampling variability, meaning their results are more spread out and more likely to be far from the true value. Researcher A, with only 20 trials, has the smallest sample size and is therefore most likely to obtain a result that deviates significantly from 0.6.

Question 15

A botanist knows that the probability of a certain plant producing a flower with a desirable trait is 0.20. She wants to simulate the number of plants she must inspect to find the first one with the trait. Using a random number generator that produces integers from 1 to 10, which of the following describes a valid setup for a single trial?

  1. Let integers 1 and 2 represent success and 3-10 represent failure. Generate random integers until a 1 or 2 appears. The total number of integers generated is the result for the trial. (correct answer)
  2. Let integers 1 and 2 represent success and 3-10 represent failure. Generate a sample of 10 random integers. Count how many of the integers are a 1 or a 2.
  3. Let the integer 2 represent success and all other integers represent failure. Generate integers until a 2 appears. The total number of integers generated is the result for the trial.
  4. Let integer 1 represent success and 2-5 represent failure. Ignore integers 6-10. Generate random integers from the valid set, and the trial ends when a 1 appears.
Explanation: The simulation needs to model a geometric setting: waiting for the first success. The probability of success is 0.20. Option A correctly assigns 2 out of 10 integers (1 and 2) to success, making P(Success) = 2/10 = 0.20. A trial correctly consists of generating numbers until the first success occurs, and the outcome is the number of trials needed. This procedure accurately models the scenario and efficiently uses all generated digits.

Question 16

A game involves rolling a pair of standard six-sided dice. A player wins on the first roll if the sum is 7. They lose if the sum is 2, 3, or 12. If they roll any other sum, they roll again until they either win (by rolling a 7) or lose (by rolling a 2, 3, or 12). A simulation is set up to estimate the probability of winning. A trial consists of generating pairs of random integers (1-6) until the game ends. Using the sequence of simulated dice rolls below, what are the outcomes of the first three trials?

Sequence of rolls: (3, 4), (5, 6), (1, 1), (6, 2), (4, 4), (5, 2), (6, 6)

  1. Win, Lose, Win (correct answer)
  2. Win, Re-roll, Lose
  3. Win, Lose, Re-roll
  4. Win, Lose, Lose
Explanation: We must trace the game for each trial. A trial ends only with a Win (sum 7) or a Loss (sum 2, 3, or 12).
  • Trial 1: The first roll is (3, 4), sum = 7. This is a Win. The trial ends.
  • Trial 2: The next roll is (5, 6), sum = 11. This is a re-roll. The next roll is (1, 1), sum = 2. This is a Loss. The trial ends.
  • Trial 3: The next roll is (6, 2), sum = 8. This is a re-roll. The next roll is (4, 4), sum = 8. This is another re-roll. The next roll is (5, 2), sum = 7. This is a Win. The trial ends. The outcomes of the first three trials are Win, Lose, Win.

Question 17

In a carnival game, a player tosses a ring at a board of 20 pegs. The player has a 30% chance of hooking a peg with any given toss. To estimate the probability of needing exactly 4 tosses to get the first success, a simulation is proposed. Which of the following correctly describes how to use a random number table to find the outcome of a single trial?

  1. Use two-digit numbers 00-99. Let 00-29 be a success. Generate 4 numbers. If exactly one of them is in the success range, the trial is a success.
  2. Use two-digit numbers 00-99. Let 00-29 be a success. Generate numbers one at a time. The trial's outcome is a 'yes' if the first success occurs on the fourth number generated.
  3. Use single digits 0-9. Let 0-2 be a success. Generate a sequence of digits. The number of digits generated until the first success is the result of the trial. Repeat many times and count how often the result is 4. (correct answer)
  4. Use single digits 0-9. Let 0-2 be a success. Generate 4 digits. The result of the trial is the number of successes found within those 4 digits.
Explanation: This is a geometric probability scenario: waiting for the first success. A trial should simulate this waiting process. Option C correctly sets up the probability of success (3 out of 10 digits = 30%) and defines a trial as generating digits until the first success occurs. The value recorded for the trial is the number of tosses needed. To find the desired probability, one would run many such trials and then calculate the proportion of trials where the recorded value was exactly 4.

Question 18

A quality control manager inspects a batch of 100 computer chips, of which 10 are known to be defective. They want to simulate the process of randomly selecting 5 chips without replacement to estimate the probability of getting at least one defective chip. The proposed simulation is: Use a random number generator to produce integers from 1 to 100. Let 1-10 represent defective chips and 11-100 represent non-defective chips. Generate 5 random numbers and record if at least one is between 1 and 10. Repeat this 200 times. What is the most significant flaw in this simulation design?

  1. The simulation generates 5 random numbers with replacement, but the physical selection is done without replacement. (correct answer)
  2. Two hundred trials is not a large enough number to get a sufficiently accurate estimate of the probability.
  3. The assignment of numbers is incorrect; it should be 1-10 for defective and 11-90 for non-defective chips.
  4. The simulation should continue generating numbers until a defective chip is found, not generate a fixed sample of 5.
Explanation: The most significant flaw is the failure to account for the 'without replacement' nature of the selection. When a chip is selected for inspection, it cannot be selected again. Standard random number generators sample with replacement, meaning the same number (representing the same chip) could be generated multiple times in one trial. A correct simulation must ensure that the 5 numbers generated for a single trial are all unique.

Question 19

A company has two production lines, A and B. Line A produces 60% of phones and has a 5% defect rate. Line B produces 40% of phones and has a 10% defect rate. A simulation is designed to estimate the probability that a randomly selected defective phone came from Line A. Which simulation plan would correctly estimate this probability?

  1. Simulate a large number of phones. For each, assign it to Line A with probability 0.6. Then, determine if it is defective with probability 0.05. The result is the proportion of all simulated phones that are defective and from Line A.
  2. Simulate selecting a defective phone by first choosing a line (60% A, 40% B) and then checking a second condition for the defect rate. Repeat many times and find the proportion of trials where the line chosen was A.
  3. Simulate a large number of phones according to their production and defect rates. Then, considering only the phones that were marked as defective, calculate the proportion that came from Line A. (correct answer)
  4. Simulate an equal number of phones from Line A and Line B. For each group, use the respective defect rates to find the number of defects. The result is (defects from A) / (total defects from A + B).
Explanation: The question asks for a conditional probability: P(from Line A | phone is defective). A direct simulation of this requires creating a large population of phones that reflects the overall production process. After simulating many phones, we must first isolate the sub-population of interest—all the defective phones. The estimated probability is then the number of defective phones from Line A divided by the total number of defective phones from both lines.

Question 20

At a holiday gift exchange with 6 participants, each person brings one gift. The gifts are placed in a pile, mixed, and then each person randomly draws one gift. A simulation is designed to estimate the probability that at least one person draws their own gift. Which of the following procedures correctly models one trial of this simulation?

  1. Roll a 6-sided die 6 times. Let the sequence represent the gifts drawn by persons 1 through 6. Check if any number matches its position in the sequence (e.g., a '1' on the first roll).
  2. Create a list of numbers 1, 2, 3, 4, 5, 6. Randomly select a number from the list 6 times, with replacement, and record if any number is selected more than once.
  3. Take 6 index cards and number them 1 through 6. Shuffle the cards and arrange them in a line. Count the number of cards for which the card's number matches its position in the line (e.g., card '3' is in the 3rd position). (correct answer)
  4. Roll a 6-sided die. If it's a 1, count it as a success for one person drawing their own gift. Repeat 6 times and divide the number of successes by 6.
Explanation: This problem is a classic derangement or matching problem. Each person must draw one unique gift, meaning the process is without replacement. Procedure C correctly models this. Numbering the people and their corresponding gifts from 1 to 6, shuffling the numbered cards is equivalent to randomly distributing the gifts. Checking if a card's number matches its position is equivalent to checking if a person drew their own gift. This entire process constitutes one valid trial.