Math 1 Quiz: Simulating Probabilities

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 1.

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.

Question 1

1. The student incorrectly calculated the theoretical probability; the true value should be 75%, making the simulation result an overestimate
2. The student failed to account for simulation variability; 86.3% is reasonably close to 87.5% and represents normal sampling variation (correct answer)
3. The student misinterpreted the simulation setup; three coin flips cannot properly model the probability of having boys and girls in a family
4. The student used an insufficient number of trials; at least 10,000 trials are required before comparing simulation results to theoretical values

Question 2

1. The simulation fails to account for recovery and immunity, making it unsuitable for modeling realistic disease progression over time
2. The 15% initial infection rate is too low to generate meaningful transmission dynamics in a grid-based spatial model
3. The probability value of 0.3 for transmission is arbitrary and not based on empirical data from actual disease studies
4. The simulation uses a fixed grid structure that doesn't reflect the complex social networks and movement patterns in real populations (correct answer)

Question 3

1. Student A's simulation is more accurate because dice provide truly random outcomes, while computer generators produce only pseudo-random sequences
2. Student B's simulation is more accurate because it provides a more precise representation of the 30% probability than the 3/10 approximation
3. Both simulations are equally valid since they both correctly represent the 30% probability, though they use different random mechanisms (correct answer)
4. Student A's simulation is flawed because physical dice cannot adequately model the independence assumption required for traffic light probabilities

Question 4

1. Generating random numbers until exactly 2 left-handed students are found, then recording how many total students were needed
2. Generating exactly 4 random numbers, then checking whether exactly 2 of them fall in the range 1-12 (correct answer)
3. Generating 100 random numbers and counting how many times exactly 2 consecutive numbers fall in the range 1-12
4. Generating random numbers in groups of 2 until both represent left-handed students, then recording the number of groups needed

Question 5

1. Use digits 0-3 for cured patients and 4-9 for non-cured patients, which maintains the approximate 40% cure rate
2. Generate random single digits until exactly 40% represent cured patients, discarding any excess digits that don't fit the target percentage
3. Use digits 0-4 for cured patients and 5-9 for non-cured patients, then adjust results mathematically to account for the 50% vs 40% difference
4. Read pairs of single digits to create two-digit numbers from 00-99, then apply the original 00-39 classification scheme (correct answer)

Question 6

1. The probability of rain increases each day if no rain occurred the previous day, following a compensatory weather pattern
2. Each day has a 25% probability of rain, and the weather on different days occurs independently of each other (correct answer)
3. Rain events follow a cyclical pattern where exactly 25% of all days in any extended period will have precipitation
4. The weather system has memory, where rainy days tend to cluster together and dry days tend to cluster together

Question 7

1. Calculate the average of all 50 trial proportions, since this eliminates the random variation present in individual trials
2. Use the trial that produced results closest to 8%, since this trial best represents the theoretical probability built into the simulation
3. Count the total number of defective items across all trials and divide by the total number of items tested across all trials (correct answer)
4. Identify the median proportion from the 50 trials, since this value is less affected by extreme outlier trials than the mean

Question 8

1. 6 red balls and 4 blue balls, drawing 10 balls with replacement to simulate treating 10 patients in each trial (correct answer)
2. 7 red balls and 3 blue balls, drawing 10 balls with replacement to ensure the desired outcome occurs in each trial
3. 60 red balls and 40 blue balls, drawing 10 balls without replacement to simulate the finite population of patients
4. 10 red balls and 10 blue balls, drawing until 7 red balls are obtained to simulate achieving the target success rate

Question 9

1. Between 30 and 38 green sections, representing the expected value plus or minus roughly 2 standard deviations (correct answer)
2. Between 32 and 36 green sections, capturing the central 85% of possible outcomes around the expected value
3. Exactly 34 green sections in most trials, since simulations should closely match the theoretical probability
4. Between 28 and 40 green sections, since results cannot exceed the total sample size of 40

Question 10

1. The student incorrectly assumes that 20 attempts with 1/50 probability each should give a 40% chance of winning at least once (correct answer)
2. The student failed to account for the possibility of multiple wins within a single trial, which would increase the success rate
3. The student used the wrong simulation setup; random integers 1-50 with only 1 as a win represents a 2% win rate, not 1/50
4. The student's calculation is correct, indicating a fundamental error in either the simulation design or the execution of the trials

Question 11

1. Using digits 0-9 creates groups of unequal probability since there are 10 digits but only 2 outcomes needed
2. The simulation should use digits 0, 1, and 2 to represent defective items to properly model 15% probability
3. 200 simulations is too few trials to produce reliable results for this type of probability estimation
4. The random digit assignment gives 20% probability for defective items instead of the required 15% (correct answer)

Question 12

1. The concern is valid because the probability should be much lower, around 25-30%
2. The concern is unfounded because 271/500 = 54.2%, which is reasonable for this scenario (correct answer)
3. The simulation is flawed because it should use days of the year, not just months
4. The result indicates a programming error because exactly 2 people sharing should occur about 50% of the time

Question 13

1. $10 \times \frac{1}{50} = 0.20 = 20\%$
2. $1 - \left(\frac{49}{50}\right)^{10} \approx 18.3\%$ (correct answer)
3. $\binom{10}{1} \times \frac{1}{50} \times \left(\frac{49}{50}\right)^9 \approx 16.7\%$
4. $\frac{362}{2000} \times \frac{50}{10} = 90.5\%$

Question 14

1. Ignore repeated numbers since they represent different physical cards in the deck
2. Discard the entire trial and start over whenever any number repeats
3. Keep the first occurrence of each number and generate new random numbers to replace repeats (correct answer)
4. Continue the simulation as planned since repetition doesn't affect the final probability calculation

Question 15

1. The student failed to account for the without-replacement nature of the quiz questions
2. The student assumed the simulation should show exactly 4 consecutive trues, not at least 4
3. The student incorrectly calculated that 4 out of 10 questions should be true 40% of the time
4. The student confused the probability of individual events with the probability of sequential patterns (correct answer)

Question 16

1. Increase the number of trials from 1200 to 5000 while keeping the same simulation design (correct answer)
2. Change to random integers 1-10 where 1-4 represent wins, keeping 1200 trials
3. Run multiple sets of 1200 trials and average the results from each set
4. Modify the simulation to test 10 games instead of 5 to get more data per trial

Question 17

1. The claim is likely false because 32/50 = 64%, which is less than 70%
2. The claim is supported because making 14+ shots occurs in 64% of simulations under the 70% assumption (correct answer)
3. The simulation is invalid because 50 trials is too small a sample size to draw conclusions
4. The claim is false because the player should make exactly 14 shots if she truly shoots 70%

Question 18

1. Simulation 1 is more accurate because it uses a larger range of random numbers
2. Simulation 2 is more accurate because 3/25 = 12% exactly matches the given probability
3. Both simulations are properly designed and the results show reasonable agreement (correct answer)
4. The results suggest an error because both simulations should converge to exactly the same value

Question 19

1. Both simulations are correctly designed, and the small difference is due to random variation in sampling (correct answer)
2. Student A's design is flawed because using 0s and 1s doesn't properly simulate fair coin flips
3. Student B's design is superior because counting occurrences is more accurate than using random numbers
4. The difference in results indicates that one student made an error in counting or recording data

Question 20

1. Generate random decimals from 0 to 1; values ≤ 0.60 represent success, then count successes in groups of 10 (correct answer)
2. Generate random integers from 1 to 10; values ≤ 7 represent success, then count successes in groups of 10
3. Generate random integers from 1 to 5; values 1 and 2 represent success, then count successes in groups of 10
4. Generate random integers from 0 to 9; values 0-4 represent success, then count successes in groups of 10