Probability and Statistics Test Review – Step-by-Step Guidance

Q1. Which among the following numbers could be the probability of an event? 0.23, 0, 3/2, 3/4, −1.32

Background

Topic: Probability Values

This question tests your understanding of the range of valid probability values for any event.

Key Terms and Formulas

• Probability: A measure of how likely an event is to occur, always between 0 and 1 (inclusive).

• Valid Probability Range: $0 \leq P(E) \leq 1$

Step-by-Step Guidance

1. Recall that a probability must be a number between 0 and 1, inclusive.

2. Check each number given: Is it within the interval $[0, 1]$?

3. Eliminate any numbers less than 0 or greater than 1.

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Q2. The probability that event A will occur is P(A) = Number of successful outcomes / Number of unsuccessful outcomes. True or False?

Background

Topic: Definition of Probability

This question checks your understanding of the correct formula for probability.

Key Terms and Formulas

• Probability of an event: $P(A) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}$

Step-by-Step Guidance

1. Recall the standard formula for probability.

2. Compare the formula given in the question to the standard formula.

3. Decide if the formula in the question is correct or not, and explain why.

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Q3. A fair coin is tossed two times in succession. The set of equally likely outcomes is {HH, HT, TH, TT}. Find the probability of getting the same outcome on each toss. (Write your answer as a reduced fraction.)

Background

Topic: Basic Probability – Sample Space

This question tests your ability to count favorable outcomes and calculate probability for simple experiments.

Key Terms and Formulas

• Sample Space: The set of all possible outcomes.

• Probability: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Step-by-Step Guidance

1. List all possible outcomes for two coin tosses: {HH, HT, TH, TT}.

2. Identify which outcomes represent "the same outcome on each toss" (i.e., both heads or both tails).

3. Count the number of favorable outcomes and the total number of outcomes.

4. Set up the probability as a fraction and reduce it if possible.

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Q4. Classify the statement as an example of classical probability, empirical probability, or subjective probability: "It is known that the probability of hitting a pothole while driving on a certain road is 1%."

Background

Topic: Types of Probability

This question tests your understanding of the three main types of probability: classical, empirical, and subjective.

Key Terms and Definitions

• Classical Probability: Based on equally likely outcomes (theoretical).

• Empirical Probability: Based on observed data or experiments.

• Subjective Probability: Based on personal judgment or opinion.

Step-by-Step Guidance

1. Consider whether the probability is based on theory, data, or opinion.

2. Decide which type of probability best fits the context of the statement.

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Q5. Suppose P(E) = 0.3, P(F) = 0.5, and P(E or F) = 0.7. Are the events E and F mutually exclusive? Explain.

Background

Topic: Mutually Exclusive Events and Addition Rule

This question tests your understanding of mutually exclusive events and how to use the addition rule for probabilities.

Key Terms and Formulas

• Mutually Exclusive Events: Events that cannot happen at the same time (no overlap).

• Addition Rule: $P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)$

Step-by-Step Guidance

1. Recall that for mutually exclusive events, $P(E \text{ and } F) = 0$.

2. Use the addition rule to solve for $P(E \text{ and } F)$ using the given probabilities.

3. Compare your result to the definition of mutually exclusive events.

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Q6. Suppose P(E) = 0.3, P(F) = 0.5, and P(E and F) = 0. Are the events E and F mutually exclusive? Explain.

Background

Topic: Mutually Exclusive Events

This question checks your understanding of the definition of mutually exclusive events using probability values.

Key Terms and Formulas

• Mutually Exclusive Events: $P(E \text{ and } F) = 0$

Step-by-Step Guidance

1. Recall the definition of mutually exclusive events.

2. Compare the given value of $P(E \text{ and } F)$ to the definition.

3. Decide if the events are mutually exclusive and explain your reasoning.

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Q7. The table below describes the exercise habits of a group of people with high blood pressure. If a person is selected at random, find the probability of getting someone who exercises occasionally or is a man. (Round your answer to three decimal places.)

Background

Topic: Probability with Contingency Tables (Two-Way Tables)

This question tests your ability to use a contingency table to find probabilities involving "or" (union) events.

Key Terms and Formulas

• Contingency Table: A table showing the frequency distribution of variables.

• Addition Rule for Probability: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Step-by-Step Guidance

1. Find the total number of people in the group (sum of all entries).

2. Find the number of people who exercise occasionally.

3. Find the number of men in the group.

4. Find the number of men who exercise occasionally (the overlap).

5. Apply the addition rule to avoid double-counting the overlap.

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Q8. A game has three outcomes. The probability of a win is 0.4, the probability of tie is 0.5, and the probability of a loss is 0.1. In a single play, what is the probability of not winning?

Background

Topic: Complementary Events

This question tests your understanding of how to find the probability of the complement of an event.

Key Terms and Formulas

• Complement: The event that the original event does not occur.

• Complement Rule: $P(\text{not } A) = 1 - P(A)$

Step-by-Step Guidance

1. Identify the probability of a win ($P(\text{win}) = 0.4$).

2. Use the complement rule to find the probability of not winning.

3. Alternatively, add the probabilities of the other outcomes (tie and loss).

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Q9. You are playing roulette at a casino in the United States. The wheel has 18 red slots, 18 black slots, and two green slots. In 4 spins, what is the probability of at least one red? (Round to the nearest ten-thousandth.)

Background

Topic: Probability of "At Least One" Using Complements

This question tests your ability to use the complement rule and independent events in probability.

Key Terms and Formulas

• Complement Rule: $P(\text{at least one}) = 1 - P(\text{none})$

• Probability of Red in One Spin: $P(\text{red}) = \frac{18}{38}$

• Probability of No Red in Four Spins: $P(\text{no red in 4 spins}) = (P(\text{not red}))^4$

Step-by-Step Guidance

1. Calculate the probability of not getting red in one spin: $P(\text{not red}) = 1 - P(\text{red})$.

2. Raise this probability to the 4th power to get the probability of no red in 4 spins.

3. Subtract this result from 1 to get the probability of at least one red in 4 spins.

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Q10. A car was randomly selected from a used car lot. Given that the car selected was over 10 years old, what is the probability that it was a domestic car?

Background

Topic: Conditional Probability

This question tests your ability to use conditional probability with a contingency table.

Key Terms and Formulas

• Conditional Probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Given: The car is over 10 years old (so restrict your sample space to that group).

Step-by-Step Guidance

1. Find the total number of cars over 10 years old (sum the appropriate column in the table).

2. Find the number of domestic cars over 10 years old.

3. Set up the conditional probability as the ratio of domestic cars over 10 years old to all cars over 10 years old.

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Q11. A bag contains 10 white, 12 blue, 13 red, 7 yellow, and 8 green wooden balls. A ball is selected and kept. Then a second ball is drawn and kept. What is the probability of selecting two red balls? (Round to the nearest thousandth.)

Background

Topic: Probability Without Replacement (Dependent Events)

This question tests your understanding of dependent events and how to calculate probabilities without replacement.

Key Terms and Formulas

• Probability of Two Dependent Events: $P(A \text{ and } B) = P(A) \times P(B|A)$

• Total Number of Balls: Add up all the balls in the bag.

Step-by-Step Guidance

1. Calculate the total number of balls in the bag.

2. Find the probability of drawing a red ball first.

3. After removing one red ball, find the probability of drawing a second red ball.

4. Multiply the two probabilities to get the probability of both events happening in sequence.

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Q12. The number of goals scored in a hockey game: Is this random variable discrete or continuous?

Background

Topic: Types of Random Variables

This question tests your ability to distinguish between discrete and continuous random variables.

Key Terms

• Discrete Random Variable: Takes on countable values (e.g., 0, 1, 2, ...).

• Continuous Random Variable: Takes on any value in an interval (e.g., height, weight).

Step-by-Step Guidance

1. Ask yourself: Can the number of goals be counted in whole numbers?

2. Decide if the variable is discrete or continuous based on its possible values.

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Q13. The speed of a car on a New York tollway during rush hour traffic: Is this random variable discrete or continuous?

Background

Topic: Types of Random Variables

This question tests your understanding of continuous versus discrete variables.

Key Terms

• Discrete Random Variable: Countable values.

• Continuous Random Variable: Any value within a range.

Step-by-Step Guidance

1. Consider whether speed can take on any value within a range or only specific values.

2. Classify the variable as discrete or continuous.

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Q14. Explain why the given table of probabilities for the random variable x forms a probability distribution.

Background

Topic: Probability Distributions

This question tests your understanding of the requirements for a valid probability distribution.

Key Terms and Criteria

• Each probability $P(x)$ must be between 0 and 1, inclusive.

• The sum of all probabilities must be 1: $\sum P(x) = 1$

Step-by-Step Guidance

1. Check that each $P(x)$ is between 0 and 1.

2. Add up all the $P(x)$ values to see if they sum to 1.

3. If both conditions are met, explain why this is a valid probability distribution.

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Q15. Determine the mean for this discrete probability distribution (germination data table).

Background

Topic: Mean of a Discrete Probability Distribution

This question tests your ability to calculate the expected value (mean) for a discrete probability distribution.

Key Terms and Formulas

• Mean (Expected Value): $\mu = \sum x \cdot P(x)$

Step-by-Step Guidance

1. Multiply each value of $x$ by its corresponding probability $P(x)$.

2. Add up all these products to get the mean.

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Q16. Determine the standard deviation for this discrete probability distribution (germination data table).

Background

Topic: Standard Deviation of a Discrete Probability Distribution

This question tests your ability to calculate the standard deviation for a discrete probability distribution.

Key Terms and Formulas

• Standard Deviation: $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$

• Use the mean $\mu$ you found in the previous question.

Step-by-Step Guidance

1. For each $x$, subtract the mean $\mu$ and square the result.

2. Multiply each squared difference by its corresponding $P(x)$.

3. Add up all these products.

4. Take the square root of the sum to get the standard deviation.

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Q17. Decide whether the experiment is a binomial experiment and explain why: Selecting five cards, one at a time without replacement, from a standard deck. The random variable is the number of picture cards obtained.

Background

Topic: Binomial Experiments

This question tests your understanding of the criteria for a binomial experiment.

Key Terms and Criteria

• Binomial Experiment: Must meet these conditions:

  • Fixed number of trials

  • Each trial is independent

  • Each trial has two possible outcomes (success/failure)

  • Probability of success is the same for each trial

Step-by-Step Guidance

1. Check if the number of trials is fixed (5 cards drawn).

2. Determine if each draw is independent (does the probability change after each draw?).

3. Check if there are only two outcomes per trial (picture card or not).

4. Decide if the probability of success remains constant for each draw.

5. Based on your analysis, decide if this is a binomial experiment and explain why or why not.

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Q18. The probability that a house in an urban area will develop a leak is 4%. If 39 houses are randomly selected, what is the probability that none of the houses will develop a leak? (Round to the nearest thousandth.)

Background

Topic: Binomial Probability (Probability of Zero Successes)

This question tests your ability to use the binomial probability formula for zero successes.

Key Terms and Formulas

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Here, $n = 39$, $k = 0$, $p = 0.04$

Step-by-Step Guidance

1. Identify $n$, $k$, and $p$ for this problem.

2. Plug these values into the binomial formula for $k = 0$.

3. Simplify the formula as much as possible before calculating.

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Q19. A quiz consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass the quiz a student must get 60% or better. If a student randomly guesses, what is the probability that the student will pass the quiz? (Round to the nearest thousandth.)

Background

Topic: Binomial Probability (Cumulative Probability)

This question tests your ability to use the binomial distribution to find the probability of getting at least a certain number of successes.

Key Terms and Formulas

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Here, $n = 10$, $p = 0.2$ (since 1 out of 5 answers is correct), and you need $k \geq 6$.

Step-by-Step Guidance

1. Determine the minimum number of correct answers needed to pass (60% of 10 questions).

2. Set up the sum of binomial probabilities for $k = 6, 7, 8, 9, 10$.

3. Write out the binomial formula for each value of $k$ and add them together.

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Q20. According to the FCC, 70% of all U.S. households have VCRs. In a random sample of 15 households, what is the probability that the number of households with VCRs is between 10 and 12, inclusive? (Round to the nearest thousandth.)

Background

Topic: Binomial Probability (Range of Successes)

This question tests your ability to calculate the probability of a range of successes in a binomial experiment.

Key Terms and Formulas

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Here, $n = 15$, $p = 0.7$, and $k = 10, 11, 12$.

Step-by-Step Guidance

1. Set up the binomial probability formula for $k = 10$, $k = 11$, and $k = 12$.

2. Add the probabilities for these three values to get the total probability.

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Q21. What is the mean of the number of adults that were never in a museum? (Given: probability is 20%, sample size is 70)

Background

Topic: Mean of a Binomial Distribution

This question tests your ability to find the expected value (mean) for a binomial distribution.

Key Terms and Formulas

• Mean of Binomial Distribution: $\mu = n p$

• Here, $n = 70$, $p = 0.2$

Step-by-Step Guidance

1. Identify the values of $n$ and $p$.

2. Multiply $n$ by $p$ to find the mean.

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Q22. What is the standard deviation of the number of adults that were never in a museum? (Given: probability is 20%, sample size is 70)

Background

Topic: Standard Deviation of a Binomial Distribution

This question tests your ability to calculate the standard deviation for a binomial distribution.

Key Terms and Formulas

• Standard Deviation of Binomial Distribution: $\sigma = \sqrt{n p (1-p)}$

• Here, $n = 70$, $p = 0.2$

Step-by-Step Guidance

1. Identify the values of $n$ and $p$.

2. Plug these values into the formula $\sigma = \sqrt{n p (1-p)}$.

3. Simplify the expression as much as possible before calculating.

Try solving on your own before revealing the answer!