Introductory Statistics Probability and Independence Review Guidance

Q1a. If one candy is randomly chosen, what is the probability that the chosen candy is a nut candy?

Background

Topic: Basic Probability

This question tests your understanding of calculating the probability of a single event from a finite sample space.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Count the total number of candies: chocolates, nut, and caramels.

2. Identify the number of nut candies.

3. Set up the probability formula using the numbers from steps 1 and 2.

Try solving on your own before revealing the answer!

Final Answer: 0.324

$P(\text{nut candy}) = \frac{12}{37} \approx 0.324$

There are 12 nut candies out of 37 total candies.

Q1b. If one candy is randomly chosen, what is the probability that the chosen candy is not a nut candy?

Background

Topic: Complement Rule in Probability

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

Key Terms and Formulas

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

Step-by-Step Guidance

1. Recall the probability of choosing a nut candy from Q1a.

2. Apply the complement rule to find the probability of not choosing a nut candy.

Try solving on your own before revealing the answer!

Final Answer: 0.676

$P(\text{not nut candy}) = 1 - 0.324 = 0.676$

The probability of not picking a nut candy is the complement of picking a nut candy.

Q1c. If two candies are chosen, without replacement, what is the probability that they are both caramels?

Background

Topic: Probability of Compound Events (Without Replacement)

This question tests your understanding of dependent events and how probabilities change when sampling without replacement.

Key Terms and Formulas

• Probability of both events: $P(A \text{ and } B) = P(A) \times P(B|A)$

• Without replacement: The total number of items decreases after the first selection.

Step-by-Step Guidance

1. Find the probability of picking a caramel candy first.

2. After one caramel is removed, find the probability of picking another caramel from the remaining candies.

3. Multiply the probabilities from steps 1 and 2 to get the probability of both events occurring.

Try solving on your own before revealing the answer!

Final Answer: 0.068

$P(\text{both caramels}) = \frac{10}{37} \times \frac{9}{36} \approx 0.068$

We multiply the probability of picking a caramel first and then another caramel, accounting for the reduced total.

Q2a. How many residents participated in this poll?

Background

Topic: Data Summation

This question tests your ability to sum values from a contingency table.

Key Terms and Formulas

• Total participants: Add all values from the table.

Step-by-Step Guidance

1. Add up all the numbers in the table for each response and party.

Try solving on your own before revealing the answer!

Final Answer: 540 residents

Sum all the values in the table to get the total number of respondents.

Q2b. If one person is randomly chosen, what is the probability that they said “yes?”

Background

Topic: Probability from a Contingency Table

This question tests your ability to find the probability of a specific response from a table.

Key Terms and Formulas

• Probability: $P(A) = \frac{\text{Number who said "yes"}}{\text{Total participants}}$

Step-by-Step Guidance

1. Add up the number of people who said "yes" across all parties.

2. Divide this sum by the total number of participants.

Try solving on your own before revealing the answer!

Final Answer: 0.569

$P(\text{yes}) = \frac{137+80+75+15}{540} = \frac{307}{540} \approx 0.569$

307 people said "yes" out of 540 total participants.

Q2c. If one person is randomly chosen, what is the probability that they were an Independent?

Background

Topic: Probability from a Contingency Table

This question tests your ability to find the probability of a specific category from a table.

Key Terms and Formulas

• Probability: $P(A) = \frac{\text{Number of Independents}}{\text{Total participants}}$

Step-by-Step Guidance

1. Add up the number of Independents (all responses).

2. Divide this sum by the total number of participants.

Try solving on your own before revealing the answer!

Final Answer: 0.204

$P(\text{Independent}) = \frac{75+30+5}{540} = \frac{110}{540} \approx 0.204$

There are 110 Independents out of 540 total participants.

Q2d. If one person is randomly chosen, what is the probability that they are a Republican and said “yes?”

Background

Topic: Joint Probability

This question tests your ability to find the probability of two events occurring together (intersection).

Key Terms and Formulas

• Joint Probability: $P(A \text{ and } B) = \frac{\text{Number of Republicans who said "yes"}}{\text{Total participants}}$

Step-by-Step Guidance

1. Find the number of Republicans who said "yes" from the table.

2. Divide this number by the total number of participants.

Try solving on your own before revealing the answer!

Final Answer: 0.148

$P(\text{Republican and yes}) = \frac{80}{540} \approx 0.148$

80 Republicans said "yes" out of 540 total participants.

Q2e. Is it unusual for a respondent to be a Republican and say “yes”? Why or why not?

Background

Topic: Unusual Events in Probability

This question tests your understanding of what makes an event "unusual" in probability, typically when $P < 0.05$.

Key Terms and Formulas

• Unusual event: Probability less than 0.05 is often considered unusual.

Step-by-Step Guidance

1. Recall the probability calculated in Q2d.

2. Compare this probability to the threshold for "unusual" events (0.05).

Try solving on your own before revealing the answer!

Final Answer: No, it is not unusual.

$P(\text{Republican and yes}) = 0.148$ which is greater than 0.05, so it is not considered unusual.

Q2f. Is it unusual for a respondent to be an Independent and say “not sure?” Why or why not?

Background

Topic: Unusual Events in Probability

This question tests your understanding of the threshold for unusual events.

Key Terms and Formulas

• Unusual event: Probability less than 0.05 is often considered unusual.

Step-by-Step Guidance

1. Find the number of Independents who said "not sure" from the table.

2. Divide this number by the total number of participants.

3. Compare the result to 0.05 to determine if it is unusual.

Try solving on your own before revealing the answer!

Final Answer: Yes, it is unusual.

$P(\text{Independent and not sure}) = \frac{5}{540} \approx 0.009$ which is less than 0.05, so it is considered unusual.

Q2g. If one person is randomly chosen, what is the probability that they are a Democrat or said “yes?”

Background

Topic: Probability of Union of Two Events

This question tests your ability to find the probability of either of two events occurring, using the addition rule.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Find the probability of being a Democrat.

2. Find the probability of saying "yes."

3. Find the probability of being a Democrat and saying "yes."

4. Apply the addition rule to combine these probabilities.

Try solving on your own before revealing the answer!

Final Answer: 0.685

$P(\text{Democrat or yes}) = P(\text{Democrat}) + P(\text{yes}) - P(\text{Democrat and yes}) = \frac{200}{540} + \frac{307}{540} - \frac{137}{540} = \frac{370}{540} \approx 0.685$

We use the addition rule to avoid double-counting those who are both Democrats and said "yes."

Q3a. What is the probability of P(E and F) if E and F are mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of the definition of mutually exclusive events.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

• $P(E \text{ and } F) = 0$ if E and F are mutually exclusive.

Step-by-Step Guidance

1. Recall the definition of mutually exclusive events.

2. Apply the rule that the probability of both occurring is zero.

Try solving on your own before revealing the answer!

Final Answer: 0

Mutually exclusive events cannot happen together, so $P(E \text{ and } F) = 0$.

Q3b. What is the probability of P(E or F) if E and F are mutually exclusive?

Background

Topic: Addition Rule for Mutually Exclusive Events

This question tests your ability to use the addition rule for mutually exclusive events.

Key Terms and Formulas

• Addition Rule: $P(E \text{ or } F) = P(E) + P(F)$ for mutually exclusive events.

Step-by-Step Guidance

1. Recall the probabilities of E and F.

2. Add the probabilities together.

Try solving on your own before revealing the answer!

Final Answer: 0.65

$P(E \text{ or } F) = 0.45 + 0.20 = 0.65$

For mutually exclusive events, simply add their probabilities.

Q3c. What is the probability of P(E and F) if E and F are independent?

Background

Topic: Independent Events

This question tests your understanding of the multiplication rule for independent events.

Key Terms and Formulas

• Multiplication Rule: $P(E \text{ and } F) = P(E) \times P(F)$ for independent events.

Step-by-Step Guidance

1. Recall the probabilities of E and F.

2. Multiply the probabilities together.

Try solving on your own before revealing the answer!

Final Answer: 0.09

$P(E \text{ and } F) = 0.45 \times 0.20 = 0.09$

For independent events, multiply their probabilities.

Q3d. What is the probability of P(E or F) if E and F are independent?

Background

Topic: Addition Rule for General Events

This question tests your ability to use the general addition rule for events that may overlap.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Recall the probabilities of E and F.

2. Calculate $P(E \text{ and } F)$ using the multiplication rule for independent events.

3. Apply the addition rule to combine the probabilities.

Try solving on your own before revealing the answer!

Final Answer: 0.56

$P(E \text{ or } F) = 0.45 + 0.20 - 0.09 = 0.56$

Subtract the overlap to avoid double-counting.

Q4a. Interpret: The probability that a household in Santa Monica owns a dog is 0.38.

Background

Topic: Probability Interpretation

This question tests your ability to interpret a probability in context.

Key Terms and Formulas

• Probability as expected frequency: Multiply probability by sample size to estimate expected number.

Step-by-Step Guidance

1. Think about what the probability means in terms of expected outcomes.

2. Apply the probability to a hypothetical sample size (e.g., 100 households).

Try solving on your own before revealing the answer!

Final Answer: If we looked at 100 households in Santa Monica, we’d expect to find that 38 owned a dog.

Probability can be interpreted as the expected proportion in a large sample.

Q4b. Interpret: The probability that an oil well successfully produces oil is 0.034.

Background

Topic: Probability Interpretation

This question tests your ability to interpret a probability in context.

Key Terms and Formulas

• Probability as expected frequency: Multiply probability by sample size to estimate expected number.

Step-by-Step Guidance

1. Think about what the probability means in terms of expected outcomes.

2. Apply the probability to a hypothetical sample size (e.g., 1000 oil wells).

Try solving on your own before revealing the answer!

Final Answer: If we looked at 1000 oil wells, we’d expect 34 of them to successfully produce oil.

Probability can be interpreted as the expected proportion in a large sample.

Q5a. Are the events "A person is driving a car" and "the same person is riding a horse" mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of whether two events can occur at the same time.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

Step-by-Step Guidance

1. Consider whether a person can drive a car and ride a horse at the same time.

Try solving on your own before revealing the answer!

Final Answer: Yes, they are mutually exclusive.

A person cannot do both at the same time.

Q5b. Are the events "Roll a die and getting both an even number and a number greater than 3" mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of whether two events can overlap.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

Step-by-Step Guidance

1. List the possible outcomes for each event.

2. Check if any outcomes satisfy both conditions.

Try solving on your own before revealing the answer!

Final Answer: No, they are not mutually exclusive.

Some numbers (e.g., 4 and 6) are both even and greater than 3.

Q5c. Are the events "Doing your homework" and "listening to the radio" mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of whether two events can occur together.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

Step-by-Step Guidance

1. Consider whether a person can do both activities at the same time.

Try solving on your own before revealing the answer!

Final Answer: No, they are not mutually exclusive.

A person can do homework while listening to the radio.

Q5d. Are the events "Rolling a die and getting an even number" and "rolling a die and getting an odd number" mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of whether two events can occur together.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

Step-by-Step Guidance

1. List the possible outcomes for each event.

2. Check if any outcome can be both even and odd.

Try solving on your own before revealing the answer!

Final Answer: Yes, they are mutually exclusive.

A die roll cannot be both even and odd at the same time.

Q5e. Are the events "A student takes a math class" and "the student takes a physics class" mutually exclusive?

Background

Topic: Mutually Exclusive Events

This question tests your understanding of whether two events can occur together.

Key Terms and Formulas

• Mutually exclusive: Events cannot occur together.

Step-by-Step Guidance

1. Consider whether a student can take both classes in the same term.

Try solving on your own before revealing the answer!

Final Answer: No, they are not mutually exclusive.

A student can take both classes.

Q6a. Are the events "A student takes a math class" and "the same student majors in physics" independent?

Background

Topic: Independent Events

This question tests your understanding of whether two events affect each other's probability.

Key Terms and Formulas

• Independent: The occurrence of one event does not affect the probability of the other.

Step-by-Step Guidance

1. Consider whether majoring in physics affects the likelihood of taking a math class.

Try solving on your own before revealing the answer!

Final Answer: No, they are not independent.

Physics majors are more likely to take math classes, so the events are not independent.

Q6b. Are the events "A coin flip comes up heads" and "a roll of a die results in an even number" independent?

Background

Topic: Independent Events

This question tests your understanding of whether two events affect each other's probability.

Key Terms and Formulas

• Independent: The occurrence of one event does not affect the probability of the other.

Step-by-Step Guidance

1. Consider whether the outcome of a coin flip affects the outcome of a die roll.

Try solving on your own before revealing the answer!

Final Answer: Yes, they are independent.

The outcome of the coin flip does not affect the die roll.

Q6c. Are the events "It rains today" and "a baseball game is cancelled today" independent?

Background

Topic: Independent Events

This question tests your understanding of whether two events affect each other's probability.

Key Terms and Formulas

• Independent: The occurrence of one event does not affect the probability of the other.

Step-by-Step Guidance

1. Consider whether rain affects the likelihood of a baseball game being cancelled.

Try solving on your own before revealing the answer!

Final Answer: No, they are not independent.

Rain increases the chance of a baseball game being cancelled.

Q6d. Are the events "It rains today" and "my calculator needs new batteries today" independent?

Background

Topic: Independent Events

This question tests your understanding of whether two events affect each other's probability.

Key Terms and Formulas

• Independent: The occurrence of one event does not affect the probability of the other.

Step-by-Step Guidance

1. Consider whether rain affects the likelihood of your calculator needing new batteries.

Try solving on your own before revealing the answer!

Final Answer: Yes, they are independent.

Rain and calculator battery life are unrelated events.