Design Simulations for Compound Events - 7th Grade Math

1. A student designs a simulation for the compound event: choose 3 library books and exactly 1 is a graphic novel. About 40% of the library books are graphic novels. Which mapping correctly represents the 40% chance each time a book is chosen (in the simulation)?
   Answer: Use a 10-section spinner: 4 sections labeled G and 6 sections labeled N. Since about 40% of library books are graphic novels, the simulation device needs 4 out of 10 outcomes to represent a graphic novel, such as a 10-section spinner with 4 sections labeled G and 6 labeled N. This matches the 40% chance exactly, since 4/10 = 0.4. Choice A reverses the spinner, using 6 sections for G and 4 for N, which represents 60% instead of 40%. Choice C uses a die with 4 out of 6 sides for graphic novel, which is about 67%, far too high. Choice D uses a coin, giving a 50% chance instead of 40%.
2. A board game has two stages in a turn. Stage 1: you draw a "bonus" card with probability $0.4$. Stage 2: you roll a 6-sided number cube and succeed if you roll a 1 (probability about $\frac{1}{6}$). You want to simulate the compound event "bonus card AND roll a 1." Which plan is best?
   Answer: Use a 10-section spinner with 4 bonus sections and a number cube; in each trial, spin once and roll once; success if bonus AND roll a 1; repeat 100 trials; estimate probability by successes/100. The situation has two independent stages, drawing a bonus card with probability 0.4 and rolling a 1 with probability about one sixth, so the simulation needs one device matched to each stage, which is exactly what choice A does with a 10 section spinner and a number cube. Choice B uses two coin flips, which does not represent either stage's probability. Choice C uses only the number cube and ignores the bonus card stage entirely, so it cannot model the compound event. Choice D uses only the spinner and ignores the number cube roll, also missing one of the two stages. Only choice A pairs a device with each stage and defines success as both events happening together in the same trial.
3. A student wants to simulate this compound event: pick 3 students; the event occurs if exactly 1 has blood type A. Assume $P(\text{type A})\approx 0.4$. Which simulation setup correctly matches the probability and the compound event?
   Answer: Use a 10-section spinner with 4 A and 6 not A. Spin 3 times per trial and count exactly 1 A as success. Repeat 100 trials. Since the probability of type A is about 0.4, the simulation device needs to match that probability, and a 10 section spinner with 4 sections labeled A does exactly that, making choice A correct when spun 3 times per trial to count exactly 1 A. Choice B uses a coin, which represents a probability of 0.5, not 0.4, so it does not match the given probability. Choice C only uses a single roll of a die and does not simulate picking 3 students at all. Choice D uses a spinner with 6 A sections instead of 4, representing a probability of 0.6 instead of 0.4. Matching both the correct probability and the correct number of trials per simulation round is essential for this type of problem.
4. A school nurse says about $40\%$ of students have blood type A. Jamal wants to simulate the chance that the first student he checks has blood type A. Which simulation plan is best?
   Answer: Use a 10-section spinner with 4 sections labeled A and 6 labeled not A; spin 100 times; estimate $P(\text{A})=\frac{\text{number of A spins}}{100}$. This question tests designing simulations using random devices to estimate compound event probabilities through trials and frequency. The design involves: (1) identifying the event as the probability that the first student checked has type A blood (P≈0.4), (2) choosing a device like a 10-section spinner with 4 sections labeled A (P=4/10=0.4), (3) defining success as landing on A, (4) running multiple trials such as 100 spins, (5) estimating P as the number of successes divided by trials. For example, with a 10-section spinner (4 A sections), after 100 spins with 42 A's, the estimate is 42/100=0.42; for expected trials to get 10 successes at P=0.4, it's about 10/0.4=25 trials. The correct design is option B, which uses a matching spinner, spins 100 times, and estimates properly for the single event of the first student having type A. Errors in other options include mismatched probabilities (coin is 0.5, cube 4/6≈0.67, cards 13/52=0.25) and insufficient trials or improper setup. When designing such simulations: (1) analyze the event and its probability, (2) select a device that matches the probability, (3) map outcomes to define success, (4) specify a sufficient number of trials, (5) describe how to estimate by frequency. Common mistakes include probability mismatches, too few trials, undefined mappings, or arithmetic errors in estimation.
5. A cafeteria line has a 40% chance that the next student chooses pizza. About how many students would you expect to check before you have found 10 pizza choices? (Use estimation.)
   Answer: About 25 students The chance of choosing pizza is 40%, or 0.4, so on average you would expect 1 pizza choice for every 2.5 students checked, since 1 divided by 0.4 equals 2.5. To reach 10 pizza choices, multiply 10 by 2.5, or equivalently divide 10 by 0.4, which gives 25 students. This matches Choice A. Choice B, about 4 students, comes from multiplying 10 by 0.4 instead of dividing, which answers a different question. Choice C, about 40 students, and Choice D, about 10 students, do not match the correct expected-value calculation for reaching 10 successes at a 40% success rate.
6. A teacher says a student has a $0.4$ chance of choosing chocolate milk on any day. The teacher wants to estimate how many days (trials) it will take to see about 10 chocolate-milk choices. About how many trials should the teacher expect?
   Answer: About 25 trials This question tests designing simulations using random devices to estimate compound event probabilities through trials and frequency, specifically calculating expected trials for about 10 successes at P=0.4. The formula for expected trials to reach k successes in Bernoulli trials is approximately k/P, so for 10 chocolate choices at P=0.4, expect 10/0.4=25 trials. Example: if simulating daily choices with P=0.4, the teacher should anticipate around 25 days to observe about 10 chocolate selections based on the inverse of probability. The correct answer is C, about 25 trials, directly from the calculation 10/0.4=25. Common errors involve wrong applications like k*P (4), 1/P (2.5 but not matching choices), or P*k (4), or misreading as total successes needed. When incorporating expected trials in designs, analyze success probability, use k/P for expectation, integrate into simulation planning for sufficient runs. Mistakes often include arithmetic errors or confusing expected value with other metrics; verify by recomputing k divided by P.
7. A basketball player makes a free throw with probability about 0.7. You want to simulate the compound event: the player takes 3 free throws and makes at least 2. Which simulation plan is best?
   Answer: Use a 10-section spinner with 7 sections labeled Make and 3 labeled Miss. For each trial, spin 3 times and record whether at least 2 are Make. Repeat 100 trials and estimate the probability by successes/trials. To simulate 3 free throws with a 70% chance of making each one, use a device where 7 out of 10 outcomes represent a Make, such as a 10-section spinner with 7 sections for Make and 3 for Miss. Each trial should model all 3 attempts by spinning 3 times, then checking whether at least 2 of those 3 spins are Makes. Running 100 trials provides enough data for a reliable estimate, matching Choice B. Choice A uses a coin, which represents only a 50% chance, not 70%, and it uses just 10 trials, far too few for a reliable estimate. Choice C spins only once and applies that single result to represent all 3 free throws, which does not model 3 independent attempts. Choice D also uses a single roll to stand in for the whole event, treating one roll of 1-4 as automatically meaning at least 2 makes, which does not model three separate attempts either.
8. A science club says a homemade rocket launches successfully about 60% of the time. You want to simulate the compound event: in 3 launches, the rocket succeeds exactly 2 times. Which plan is a complete and correct simulation?
   Answer: Use a 10-section spinner with 6 sections labeled Success and 4 labeled Fail. Spin 3 times per trial, count trials with exactly 2 Successes, repeat 100 trials, and estimate P as successes/100. To match the real probability of 60% success, use a device where 6 out of 10 outcomes represent Success, such as a 10-section spinner with 6 sections labeled Success and 4 labeled Fail. Each trial should model all 3 launches by spinning 3 times, then counting how many trials result in exactly 2 Successes out of those 3 spins. Repeating this for 100 trials gives a reliable estimate of the probability. Choice A uses a die with only 2 out of 6 sides marked Success, which is about 33%, not 60%, and it only uses 5 trials, which is too few for a solid estimate. Choice B spins only once and applies that single result to all 3 launches, which does not simulate 3 independent attempts. Choice C uses a coin, giving a 50% chance instead of 60%.
9. A game booth has a prize wheel where the chance of winning a small prize is about 0.3. You want to simulate the compound event: spin 5 times and win exactly 2 prizes. Which simulation plan is best?
   Answer: Use a 10-section spinner with 3 sections labeled Win and 7 labeled Lose. Spin 5 times per trial, record whether exactly 2 Wins occur, repeat 100 trials, and estimate P as successes/100. A win probability of about 0.3 means the device should have roughly 3 out of 10 equally likely outcomes labeled Win, matching choice D's 10-section spinner with 3 Win sections. Spinning 5 times per trial and checking for exactly 2 Wins simulates the compound event, and running 100 trials gives a stable estimate. Choice A reverses the Win/Lose split (7 Win sections would mean P=0.7, not 0.3) and also uses far too few trials (10). Choices B and C use devices with a 0.5 chance of winning (a die split evenly, or a coin), which don't match the actual 0.3 probability.
10. A basketball player makes a free throw about 70% of the time. You want to simulate the compound event: the player makes at least 3 out of 4 free throws. Which device and mapping best match this situation?
    Answer: Use a 10-section spinner with 7 sections labeled Make and 3 labeled Miss. Spin 4 times per trial and count trials with 3 or 4 Makes over 100 trials. To match a 70% chance of success, use a device where 7 out of 10 outcomes count as a Make, such as a 10-section spinner with 7 sections labeled Make and 3 labeled Miss. Each trial should model 4 free throws by spinning 4 times, then checking whether the trial has 3 or 4 Makes. Running 100 trials gives enough data for a reliable probability estimate. Choice A uses a die where only 2 out of 6 sides represent a Make, which is about 33%, not 70%. Choice C reverses the spinner so only 3 out of 10 sections are Make, giving about 30% instead of 70%. Choice D uses a coin, which gives a 50% chance instead of 70%.
11. In a game, a player has a 40% chance to win each round. Maya wants to simulate the compound event: winning exactly 2 rounds out of 3. Which outcome mapping is correct if she uses a 10-section spinner with 4 win sections and 6 lose sections?
    Answer: Spin 3 times; record a success if there are exactly 2 wins and 1 loss in any order. Winning exactly 2 out of 3 rounds means spinning 3 times per trial and checking whether exactly 2 of those spins land on win, in any order, matching choice B. Choice A doesn't make sense as written, since a single spin can't land on win twice. Choice C simulates winning all 3 rounds, which is a stricter event than exactly 2. Choice D simulates winning at least 1 round, which would count trials with 1, 2, or 3 wins, not specifically 2.
12. A school nurse knows about $40\%$ of students have blood type A. The nurse wants to simulate the compound event: selecting 10 donors and counting how many are type A. Which simulation design is best? (Your design should include a random device, how outcomes match blood types, and enough trials to estimate the probability.)
    Answer: Use a 10-section spinner with 4 sections labeled A and 6 labeled not A. For each trial, spin 10 times (one per donor), record how many A's occur, and repeat for 100 trials. Since about 40 percent of students have blood type A, the simulation device needs to match that probability, and a 10 section spinner with 4 sections labeled A does exactly that, making choice A correct when spun 10 times per trial and repeated for 100 trials. Choice B relies on picking students the nurse already knows, which is not a random method at all and would likely be biased. Choice C uses a coin, which represents a probability of 0.5, not 0.4, so it does not match the given probability. Choice D uses a die with 4 of 6 sides marking type A, representing a probability of about 0.67, another mismatch. Repeating the simulation across many trials, such as 100, rather than relying on just one round of 10 actions, is also necessary to get a reliable estimate of the probability.
13. A game booth works like this: You roll a fair die twice. You win a prize if both rolls are even (2, 4, or 6). Which simulation would correctly estimate the probability of winning?
    Answer: Roll a die twice per trial and count a win if both rolls are even; repeat 100 trials and estimate $P(\text{win})$ as wins/100. This question tests designing simulations using random devices to estimate compound event probabilities through trials and frequency, for winning by rolling two even numbers on a die (P(even)=0.5). Design: (1) identify the event as two independent rolls both even, (2) use a die (even on 2,4,6 for P=3/6=0.5), (3) define success as both even in two rolls per trial, (4) perform many trials (e.g., 100), (5) estimate P as wins over trials. Example: roll a die twice per trial, check if both are even, repeat 100 times, and calculate the proportion of wins to estimate the 0.25 probability. Choice B is correct, using the die to match the game, compounding two rolls per trial, defining the both-even success, and running 100 trials. Errors include single rolls, mismatched devices or probabilities, or insufficient trials like only two total rolls. In designing, examine the event structure, choose a matching random device, map outcomes properly, specify trial numbers, and outline frequency estimation. Run simulations by randomizing actions, recording results, counting successes, and computing estimates; avoid mismatches, low trials, incorrect mappings, or errors in calculation.
14. A board game has two stages each turn: - Stage 1: You draw a card. There is a $0.4$ chance it is a "Move" card. - Stage 2: You roll a die. There is a $\tfrac{1}{6}$ chance you roll a 6. You want to simulate the compound event: draw a Move card AND roll a 6 in the same turn. Which simulation is best?
    Answer: Use a 10-section spinner with 4 sections labeled Move and 6 labeled Not Move, then roll a die. Count a success only if the spinner lands on Move and the die shows 6. Repeat 100 turns. This question tests designing simulations using random devices to estimate compound event probabilities through trials and frequency, for the AND event of Move card (P=0.4) and rolling 6 (P=1/6). Design: (1) identify independent events, (2) use separate devices like 10-spinner (4 Move for P=0.4) and die, (3) success if both occur per trial, (4) run 100 trials, (5) estimate as successes/trials. Example: spin for Move, roll die, success only if Move and 6, repeat 100 times for frequency approximating 0.4*(1/6)≈0.067. Choice B is correct, using spinner for 0.4, die for 1/6, combining for AND, and 100 trials. Errors include single devices without compounding, mismatched P (e.g., 1-2 on die=1/3), or incorrect success definitions. Design by breaking into parts, choosing devices per probability, mapping to joint success, specifying trials, outlining estimation. Execute with independent randomizations, result tracking, counting, calculating; pitfalls are mismatches, insufficient trials, logic errors in AND, arithmetic.
15. A basketball player makes a free throw about 70% of the time. Kira wants to simulate the compound event of taking 2 free throws and making both. Which plan is best?
    Answer: Use a 10-section spinner with 7 sections labeled make and 3 labeled miss; spin twice per trial; repeat 100 trials; estimate P(both) as the number of double-make trials divided by 100. To simulate this compound event, the device used for one trial must match the free-throw probability of 70% and must represent two attempts, since Kira wants the probability of making both. A 10-section spinner with 7 make sections models 70% correctly, and spinning it twice per trial captures both free throws; running many trials and finding the fraction with two makes estimates P(both). Choice B is wrong because a coin only models a 50% probability, not 70%. Choice C is wrong because rolling one cube models a single event at about 83%, and using only one roll doesn't represent two attempts. Choice D is wrong because a 4-section spinner with 3 make sections models 75%, not 70%, and spinning only once doesn't represent the compound event of two free throws.
16. A factory estimates that about 10% of its pens are defective. You want to simulate the compound event: checking 5 pens and finding at least 1 defective. Which choice is the best simulation design using simple random devices?
    Answer: Use a 10-section spinner with 1 section labeled Defective and 9 labeled OK. For each trial, spin 5 times and record whether at least 1 Defective occurs. Repeat 100 trials and estimate with successes/trials. To simulate this compound event, you need a device where each trial matches the real probability of 10% defective. A 10-section spinner with 1 section for Defective and 9 for OK works because 1/10 equals 0.1, the same rate as the pens. Spinning it 5 times per trial models checking 5 pens, and repeating this 100 times gives enough data for a reliable estimate. Choice A uses a coin, which represents a 50% chance, not 10%, so it does not match the problem. Choice B uses a die with a 1 in 6 chance, and Choice C is not random at all since it relies on guessing which pens look defective.
17. A school nurse says about 40% of students who come in have a sore throat. You want to simulate the compound event: select 5 students and get exactly 2 with sore throats. Which simulation plan is best?
    Answer: Use a 10-section spinner with 4 sections labeled S (sore throat) and 6 labeled N. Spin 5 times per trial, record whether exactly 2 S occur, repeat 100 trials, and estimate $P$ as (number of successful trials)/100. Since about 40 percent of students have a sore throat, the simulation device needs to match that probability, and a 10 section spinner with 4 sections labeled S does exactly that, making choice C correct. Choice A uses a die with 4 of 6 faces marking sore throat, which represents a probability of about 67 percent, not 40 percent. Choice B uses a coin, representing a 50 percent probability, another mismatch. Choice D flips the spinner sections, using 6 S and 4 N, and only uses the first spin, so it neither matches the probability nor tests all 5 students per trial. Only choice C correctly models both the 40 percent probability and the 5 student trial structure.
18. A student wants to simulate the compound event: choose 4 students and at least 1 has a birthday in March. Assume the probability a randomly chosen student has a March birthday is about 1/12 (approximately 0.083). Which device and mapping is the best simple approximation?
    Answer: Use a 12-section spinner; 1 section = March birthday and 11 sections = not March birthday. The event has a probability of about 1/12, or roughly 0.083, so the best simulation device is one where a single outcome has that same chance - a 12-section spinner with 1 section marked March, matching choice A. A coin gives a 1/2 chance, far too high to represent 1/12. A die gives a 1/6 (about 0.167) chance for any one face, still much higher than 1/12. A 10-section spinner gives at best 1/10 = 0.1 per section, still not as close a match as the 12-section spinner.
19. A student wants to simulate this compound event: "Pick 2 students at random (with replacement). The first likes soccer ($P\approx 0.6$) and the second likes basketball ($P\approx 0.5$)." Which simulation design is best?
    Answer: Use a 10-section spinner with 6 sections labeled soccer and 4 not for the first student, and flip a coin for the second student (heads = basketball); one trial is spinner then coin; repeat 100 trials; count when spinner shows soccer and coin shows heads. Since the first student's probability of liking soccer is 0.6 and the second student's probability of liking basketball is 0.5, the simulation needs one device matched to each probability, which is exactly what choice B does with a 10 section spinner for soccer and a separate coin for basketball. Choice C uses a single coin flip with no connection to either probability, so it cannot represent this compound event at all. Choice A uses one roll of a number cube split 1-3 versus 4-6, which represents a 50-50 split for both events, not the 0.6 and 0.5 probabilities actually given. Choice D uses the same spinner, labeled for 0.6, for both students, which mismatches the second student's actual probability of 0.5. Matching a separate, correctly labeled device to each individual probability is what makes a simulation design valid for a compound event like this one.
20. A class simulates the compound event: in 4 tries, a student gets at least 1 correct answer by guessing. Each guess has probability $0.4$ of being correct. They run 100 trials, and 82 trials had at least 1 correct answer. Based on the simulation, what is the estimated probability of the compound event?
    Answer: $0.82$ This question tests designing simulations using random devices to estimate compound event probabilities through trials and frequency. A proper design involves: (1) identifying the event probability, here 0.4 for a correct guess, (2) choosing a matching device, (3) defining success as at least 1 correct in 4 attempts per trial, (4) running trials like 100, and (5) estimating P as successes over total trials, such as 82/100=0.82. For example, in simulating P=0.4 success, run 100 trials of 4 attempts each, count trials with at least 1 success, and if 82, estimate P=0.82. The correct estimate is choice C, which is 82/100=0.82, directly from the simulation results. Errors in other choices include incorrect calculations like 0.18 (perhaps 1-0.82) in A, the base P=0.4 in B, and invalid 1.22 in D. When designing such simulations, first analyze the compound event and its base probability, then select a device that matches it and map outcomes to define success. Finally, run sufficient trials by randomizing, record and count successes, calculate the estimate as frequency, and avoid mistakes like arithmetic errors in estimation.

Start this quiz