AP Statistics Unit 4: Probability, Random Variables, and Probability Distributions Practice Exam

Assessment for Unit 4: Probability, Random Variables, and Probability Distributions

Estimated Time: 60 minutes

Exam Details

Questions: 34 total (34 multiple choice)
Time: 60 minutes (recommended)
Topics Covered: Unit 4: Probability, Random Variables, and Probability Distributions
Format: Multiple Choice questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (34 Questions)

Select the one best answer for each question.

1. A student flips a fair coin 50 times and records the results. The student observes a sequence of 6 consecutive "Heads" in the middle of the dataset and claims that the coin must not be fair, arguing that such a distinct pattern would not occur in a truly random process. Which of the following statements best evaluates the student's claim based on statistical principles?

(A)The claim is correct; a fair coin should alternate outcomes more frequently to maintain a 50% probability.(B)The claim is correct; a streak of 6 consecutive heads is statistically impossible in a sample of 50 flips.(C)The claim is incorrect; patterns, including streaks or clusters, can and do occur in data sets generated by purely random processes.(D)The claim is incorrect; the student needs to flip the coin at least 1,000 times before any pattern can be analyzed.

2. A store manager claims that 50% of customers pay with a credit card. A cashier questions this claim after observing that 15 of the last 20 customers paid with a credit card. To investigate, the cashier uses a computer to simulate 100 trials of 20 customers, assuming the manager’s claim is true (p = 0.50). Which of the following questions is the simulation primarily designed to answer?

(A)Is the true proportion of customers who pay with a credit card exactly 0.75?(B)How likely is it to observe 15 or more customers paying with a credit card in a sample of 20, purely by chance, if the true proportion is 50%?(C)Will the next 20 customers definitely follow the pattern of the manager's claim?(D)Does the use of credit cards cause customers to spend more money?

3. Researchers are investigating whether a new fertilizer increases the growth rate of tomato plants compared to a standard fertilizer. After growing 30 plants with each fertilizer, the researchers observe that the mean height of the new fertilizer group is 5 cm greater than the standard group. Which of the following questions is most directly suggested by this observed pattern in the data?

(A)What is the maximum possible height a tomato plant can reach?(B)Is the observed difference of 5 cm large enough to indicate the new fertilizer is effective, or could this difference have occurred due to random variation?(C)Are tomato plants genetically similar to other nightshade vegetables?(D)Did the researchers make a measurement error when recording the heights?

Refer to the figure below.

4. A student rolls a six-sided die 60 times and observes that the number "6" appears 18 times. The student suspects the die might be loaded to favor the number 6. The graph below displays the results of 100 simulated trials of rolling a fair die 60 times. Based on the simulation and the student's data, which of the following is the most appropriate conclusion?

(A)The pattern in the student's data is consistent with random chance because 18 is a possible outcome in the simulation.(B)The pattern suggests the die may be loaded, because observing 18 sixes is a very rare outcome in the simulation of a fair die.(C)The pattern proves the die is fair because the simulation shows that getting a 6 is random.(D)No question can be answered because the student only performed one set of 60 rolls.

5. A statistics teacher asks students to identify which of the following scenarios describes a binomial random variable. Which of the following settings meets all the conditions (Binary, Independent, Number of trials, Probability of success) for a binomial distribution?

(A)A student deals 5 cards from a standard shuffled deck of 52 cards and counts the number of aces dealt.(B)A basketball player shoots free throws until she makes 10 shots, and we count the total number of attempts required.(C)A manufacturer selects a random sample of 20 items from a production line where 5% of items are defective, and counts the number of defective items in the sample (assuming the population is very large).(D)A researcher records the number of vehicles passing through an intersection in a one-hour interval.

6. A jar contains 5 red marbles and 5 blue marbles. A student selects a marble, records its color, and sets it aside. The student repeats this process 3 times. Let X be the number of red marbles selected. Which condition for a binomial setting is most clearly violated in this scenario?

(A)Binary: The outcomes cannot be classified as success or failure.(B)Independent: The probability of selecting a red marble changes after each draw.(C)Number: The number of trials is not fixed in advance.(D)Success: The probability of success is not between 0 and 1.

7. A fair six-sided die is rolled 12 times. Let X denote the number of times a '4' is rolled. Which of the following defines the parameters n (number of trials) and p (probability of success) for the binomial distribution modeling X?

(A)n = 12, p = 1/6(B)n = 12, p = 4/6(C)n = 6, p = 1/12(D)n = 4, p = 1/6

8. A computer program simulates 50 independent coin flips ($p=0.5$ for Heads). A student observes a pattern where the first 8 flips are all Heads. The student concludes that the random number generator must be flawed because such a clear pattern should not occur in a random process. Based on statistical principles of random variation, which of the following is the best response to the student's conclusion?

(A)The student is correct; in a truly random process, the number of Heads and Tails should balance out perfectly in every small sample.(B)The student is correct; a run of 8 Heads is statistically impossible for a fair coin.(C)The student is incorrect; patterns, including streaks, can and do occur in data generated by random variation, and a single streak does not prove non-randomness.(D)The student is incorrect; the probability of Heads increases after a streak of Heads to maintain the average.

9. A local bakery sells three types of bagels: sesame, poppy seed, and plain. Based on sales data, 55% of customers buy sesame bagels, 23% buy poppy seed bagels, and 22% buy plain bagels. To estimate the probability that the next 5 customers all buy the same type of bagel, a simulation will be conducted using a random number generator. Which of the following assignments of random integers from 00 to 99 correctly models the distribution of bagel preferences?

(A)Sesame: 00–55, Poppy Seed: 56–78, Plain: 79–99(B)Sesame: 01–55, Poppy Seed: 56–79, Plain: 80–99(C)Sesame: 00–54, Poppy Seed: 55–77, Plain: 78–99(D)Sesame: 00–54, Poppy Seed: 55–78, Plain: 79–99

10. A fair coin is flipped $n$ times. Let $\hat{p}$ represent the proportion of flips that result in heads. According to the Law of Large Numbers, which of the following best describes the behavior of $\hat{p}$ as $n$ increases?

(A)The proportion $\hat{p}$ will be exactly 0.5 for all values of $n$ greater than 100.(B)The difference between the number of heads and the number of tails will approach 0 as $n$ increases.(C)The proportion $\hat{p}$ will tend to get closer to 0.5 as $n$ increases.(D)The proportion $\hat{p}$ is more likely to be far from 0.5 when $n$ is large compared to when $n$ is small.

11. A box contains 10 numbered tickets: 6 tickets are red and 4 tickets are blue. A student wants to estimate the probability of drawing 2 red tickets in a row if the tickets are drawn **without replacement**. The student proposes the following simulation method: 1. Use a random number generator to select an integer from 1 to 10. 2. Assign integers 1–6 to represent a red ticket and 7–10 to represent a blue ticket. 3. Generate a first integer and record the color. 4. Generate a second integer and record the color. 5. If both recorded colors are red, count this as a success. 6. Repeat 100 times and calculate the proportion of successes. Which of the following best explains why this simulation method is incorrect?

(A)The assignment of integers to colors is incorrect because there are more red tickets than blue tickets.(B)The simulation models sampling with replacement, but the scenario requires sampling without replacement.(C)The simulation should have used a coin flip since there are only two colors.(D)100 trials are not enough to apply the Law of Large Numbers to estimate the probability.

12. A meteorologist states that there is a 30% probability of rain tomorrow. Which of the following is the best interpretation of this statement?

(A)It will rain for exactly 30% of the 24-hour period tomorrow.(B)In the next 10 days, it will rain on exactly 3 of them.(C)If we consider a very large number of days with atmospheric conditions similar to those expected tomorrow, about 30% of such days will result in rain.(D)Because the probability is less than 50%, it is certain that it will not rain tomorrow.

13. A fair four-sided die (tetrahedron) has faces numbered 1, 2, 3, and 4. The die is rolled twice, and the numbers on the bottom face are recorded. What is the probability that the sum of the two numbers is greater than 5?

(A)3/16(B)3/8(C)5/16(D)5/8

14. A large bag contains marbles of four different colors: Red, Blue, Green, and Yellow. The probability of drawing a Red marble is 0.35, and the probability of drawing a Blue marble is 0.25. If the probabilities of drawing a Green marble and a Yellow marble are equal, what is the probability of drawing a marble that is NOT Yellow?

(A)0.20(B)0.40(C)0.60(D)0.80

15. Events $A$ and $B$ are mutually exclusive. Which of the following statements is true regarding the joint probability of these events?

(A)$P(A \text{ and } B) = P(A) \times P(B)$(B)$P(A \text{ and } B) = 0$(C)$P(A \text{ and } B) = P(A) + P(B)$(D)$P(A \text{ and } B) = 1$

16. A standard deck of 52 playing cards contains four suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red; clubs and spades are black. Let $R$ be the event of drawing a red card, and let $F$ be the event of drawing a face card (Jack, Queen, or King). Are events $R$ and $F$ mutually exclusive?

(A)Yes, because a card cannot be both a color and a face card.(B)Yes, because the probability of drawing a red face card is zero.(C)No, because there are cards that are both red and face cards (e.g., King of Hearts).(D)No, because the sum of their probabilities is greater than 1.

17. A local weather station determines that on a specific day, the probability of rain ($R$) is 0.40, the probability of high winds ($W$) is 0.30, and the probability of experiencing rain or high winds ($R \cup W$) is 0.55. Which of the following correctly determines if events $R$ and $W$ are mutually exclusive?

(A)They are mutually exclusive because $0.40 + 0.30 \neq 0.55$.(B)They are mutually exclusive because $P(R) \times P(W) = 0.12$, which is not zero.(C)They are not mutually exclusive because $P(R \text{ and } W) = 0.15$.(D)They are not mutually exclusive because $P(R \text{ and } W) = 0.55$.

18. A fair six-sided die is rolled once. Consider the following two events: Event $A$: Rolling a number less than 3 (Outcomes: 1, 2) Event $B$: Rolling a number greater than 3 (Outcomes: 4, 5, 6) Which statement best explains why events $A$ and $B$ are mutually exclusive?

(A)The sum of their probabilities is less than 1.(B)They do not contain any of the same outcomes.(C)Rolling a number less than 3 affects the probability of rolling a number greater than 3.(D)They divide the sample space into two equal parts.

19. A high school survey asked 400 randomly selected students about their grade level and whether they plan to attend the upcoming homecoming dance. The results are summarized in the two-way table below. | | Attending | Not Attending | Total | | :--- | :---: | :---: | :---: | | **Junior** | 50 | 150 | 200 | | **Senior** | 100 | 100 | 200 | | **Total** | 150 | 250 | 400 | What is the probability that a randomly selected student is a Senior, given that the student is attending the dance?

(A)0.25(B)0.375(C)0.50(D)0.667

20. At a local coffee shop, 40% of customers order a pastry ($A$). Of the customers who order a pastry, 30% also order a cappuccino ($B$). What is the probability that a randomly selected customer orders both a pastry and a cappuccino?

(A)0.12(B)0.30(C)0.40(D)0.75

21. Let $C$ and $D$ be two events such that $P(C) = 0.60$, $P(D) = 0.50$, and $P(C \cup D) = 0.85$. What is the value of $P(C | D)$?

(A)0.25(B)0.42(C)0.50(D)0.85

22. A manufacturing plant uses two machines, Machine X and Machine Y, to produce chips. Machine X produces 60% of the chips, and Machine Y produces 40%. The defect rate for Machine X is 2%, while the defect rate for Machine Y is 5%. If a randomly selected chip is found to be defective, what is the probability that it came from Machine Y?

(A)0.020(B)0.032(C)0.400(D)0.625

23. In a large high school, 60% of students take a Mathematics course ($M$), 45% of students take a Science course ($S$), and 30% of students take both a Mathematics and a Science course. Based on these probabilities, which of the following statements is true regarding the independence and mutual exclusivity of events $M$ and $S$?

(A)Events $M$ and $S$ are mutually exclusive but not independent.(B)Events $M$ and $S$ are independent but not mutually exclusive.(C)Events $M$ and $S$ are neither independent nor mutually exclusive.(D)Events $M$ and $S$ are both independent and mutually exclusive.

24. A manufacturing plant has two safety systems, System A and System B, installed to prevent accidents. The probability that System A functions correctly during an emergency is 0.90, and the probability that System B functions correctly is 0.80. The functioning of System A is independent of the functioning of System B. What is the probability that at least one of the two systems functions correctly during an emergency?

(A)0.72(B)0.90(C)0.98(D)1.70

25. Let $J$ and $K$ be two events such that $P(J) = 0.4$, $P(K) = 0.5$, and $P(J \cup K) = 0.7$. Which of the following correctly describes the relationship between events $J$ and $K$?

(A)Events $J$ and $K$ are mutually exclusive because $P(J) + P(K) > P(J \cup K)$.(B)Events $J$ and $K$ are independent because $P(J \cap K) = P(J)P(K)$.(C)Events $J$ and $K$ are independent because $P(J \cup K) = P(J) + P(K) - P(J)P(K)$.(D)Events $J$ and $K$ are neither independent nor mutually exclusive.

26. The probability distribution for a discrete random variable $X$ is given by the function $P(X=x) = kx$ for the values $x = 1, 2, 3, 4$, where $k$ is a constant. What is the value of $P(X > 2)$?

(A)0.1(B)0.3(C)0.7(D)0.9

27. The table below displays the cumulative probability distribution for a discrete random variable $Y$. | $y$ | $P(Y \le y)$ | | :---: | :---: | | 1 | 0.15 | | 2 | 0.40 | | 3 | 0.75 | | 4 | 1.00 | Which of the following is the value of $P(Y = 3)$?

(A)0.35(B)0.40(C)0.75(D)0.85

28. A veterinarian records the number of pets $X$ owned by households in a small town. The table below displays a proposed probability distribution for the random variable $X$. | $X$ | 0 | 1 | 2 | 3 | | :---: | :---: | :---: | :---: | :---: | | $P(X)$ | 0.25 | 0.35 | 0.30 | 0.20 | Which of the following statements best explains why this is NOT a valid probability distribution?

(A)The probabilities are not increasing as $X$ increases.(B)The random variable $X$ takes on the value 0, which is not allowed for discrete distributions.(C)The sum of the probabilities is not equal to 1.(D)The values of the random variable $X$ are not between 0 and 1.

29. The probability distribution for the random variable $X$, representing the number of customer complaints received by a service center in a single hour, is given in the table below. | $X$ | 0 | 1 | 2 | 3 | | :--- | :---: | :---: | :---: | :---: | | $P(X)$ | 0.10 | 0.40 | 0.30 | 0.20 | What is the expected value of $X$?

(A)1.0(B)1.5(C)1.6(D)2.0

30. A carnival game involves spinning a wheel where a player can win money. The probability distribution of $X$, the net gain in dollars for a player in a single game, has a mean of $-1.50$ dollars. Which of the following is the best interpretation of this value?

(A)The player will lose 1.50 dollars in every game they play.(B)The most likely outcome of a single game is a loss of 1.50 dollars.(C)In a random sample of 10 games, the player will lose exactly 15.00 dollars total.(D)If the player plays the game a very large number of times, the average net gain per game will be approximately -1.50 dollars.

31. The random variable $X$ has the following probability distribution: | $X$ | 0 | 4 | | :--- | :---: | :---: | | $P(X)$ | 0.20 | 0.80 | What is the standard deviation of $X$?

(A)1.28(B)1.60(C)2.56(D)3.20

32. A local bakery sells two types of fruit pies: apple and cherry. The daily demand for apple pies is a random variable $A$ with a mean of 45 pies and a standard deviation of 5 pies. The daily demand for cherry pies is a random variable $C$ with a mean of 30 pies and a standard deviation of 4 pies. Assuming the demands for apple and cherry pies are independent, which of the following is the standard deviation of the total daily demand for these two types of pies?

(A)3.0 pies(B)6.4 pies(C)9.0 pies(D)41.0 pies

33. A logistics company uses two different routes to deliver packages to a remote town. The time it takes to travel Route 1 is a normally distributed random variable $R_1$ with a mean of 50 minutes and a standard deviation of 4 minutes. The time it takes to travel Route 2 is a normally distributed random variable $R_2$ with a mean of 45 minutes and a standard deviation of 3 minutes. The travel times for the two routes are independent. The company is interested in the difference in travel times $D = R_1 - R_2$. Which of the following describes the mean and standard deviation of the random variable $D$?

(A)Mean = 5 minutes; Standard Deviation = 1 minute(B)Mean = 5 minutes; Standard Deviation = 2.65 minutes(C)Mean = 5 minutes; Standard Deviation = 5 minutes(D)Mean = 95 minutes; Standard Deviation = 5 minutes

34. A machine at a bottling plant dispenses liquid into bottles. The amount of liquid dispensed into a single bottle is a random variable $X$ with a mean of 500 ml and a standard deviation of 2 ml. A quality control inspector selects a case containing 12 bottles chosen at random. Assuming the amount of liquid in each bottle is independent of the others, what is the standard deviation of the total amount of liquid in the 12 bottles?

(A)2.00 ml(B)6.93 ml(C)24.00 ml(D)144.00 ml

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