Q: A contingency table displays the results of a survey of 100 high school students. If 40 students drive a car and 60 do not, what is the marginal probability that a randomly selected student drives a car?

$$P(\text{Drives a car}) = \frac{40}{100} = 0.4$$

Q: In a drug trial for a new ADHD medication, 20 people in the placebo group improved, and 30 did not. What is the conditional probability that a person improved, given that they received the placebo?

$$P(\text{Improved} \,|\, \text{Placebo}) = \frac{20}{50} = 0.4$$

Q: A survey of 80 people found that 20% have blue eyes, 20% have green eyes, 30% have brown eyes, and 30% have hazel eyes. If 40% of people with blue eyes have blond hair, how many people in the survey have both blue eyes and blond hair?

$$0.2 \times 80 = 16$$ people have blue eyes; $$0.4 \times 16 = 6.4$$ \approx 6$ people have both blue eyes and blond hair.

Q: In a wedding survey, 78 guests are vegetarian, 16 are vegan, and 26 eat meat. If 10 vegetarians also eat gluten-free, what is the marginal probability that a randomly selected guest is vegetarian?

$$P(\text{Vegetarian}) = \frac{78}{120} = 0.65$$

Q: A contingency table shows that 15 students are seniors and drive a car, 25 are seniors and do not drive, 25 are not seniors and drive, and 35 are not seniors and do not drive. What is the probability that a randomly selected student is a senior or drives a car?

$$P(\text{Senior or Drives}) = \frac{15+25+25}{100} = \frac{65}{100} = 0.65$$

Q: In a drug trial, 40 people received the non-placebo and 30 of them improved. What is the probability that a person improved given they received the non-placebo?

$$P(\text{Improved} \,|\, \text{Non-Placebo}) = \frac{30}{40} = 0.75$$

Q: A survey of 100 students found that 30 are seniors, 40 drive a car, and 15 are seniors who drive a car. What is the probability that a randomly selected student is a senior given that they drive a car?

$$P(\text{Senior} \,|\, \text{Drives}) = \frac{15}{40} = 0.375$$

Question 1

A contingency table displays the results of a survey of 100 high school students. If 40 students drive a car and 60 do not, what is the marginal probability that a randomly selected student drives a car?

Accepted Answer

$$P(	ext{Drives a car}) = \frac{40}{100} = 0.4$$

Question 2

In a drug trial for a new ADHD medication, 20 people in the placebo group improved, and 30 did not. What is the conditional probability that a person improved, given that they received the placebo?

Accepted Answer

$$P(	ext{Improved} \,|\, 	ext{Placebo}) = \frac{20}{50} = 0.4$$

Question 3

Given a contingency table, which of the following best describes a joint probability?

Accepted Answer

The probability of two events occurring together, calculated as $$P(A \cap B) = \frac{n(A \cap B)}{n(	ext{Total})}$$

Question 4

A survey of 80 people found that 20% have blue eyes, 20% have green eyes, 30% have brown eyes, and 30% have hazel eyes. If 40% of people with blue eyes have blond hair, how many people in the survey have both blue eyes and blond hair?

Accepted Answer

$$0.2 	imes 80 = 16$$ people have blue eyes; $$0.4 	imes 16 = 6.4$$ \approx 6$ people have both blue eyes and blond hair.

Question 5

In a wedding survey, 78 guests are vegetarian, 16 are vegan, and 26 eat meat. If 10 vegetarians also eat gluten-free, what is the marginal probability that a randomly selected guest is vegetarian?

Accepted Answer

$$P(	ext{Vegetarian}) = \frac{78}{120} = 0.65$$

Question 6

Which of the following is the correct formula for conditional probability as shown in the study materials?

Accepted Answer

$$P(A \,|\, B) = \frac{P(A \cap B)}{P(B)}$$

Question 7

A contingency table shows that 15 students are seniors and drive a car, 25 are seniors and do not drive, 25 are not seniors and drive, and 35 are not seniors and do not drive. What is the probability that a randomly selected student is a senior or drives a car?

Accepted Answer

$$P(	ext{Senior or Drives}) = \frac{15+25+25}{100} = \frac{65}{100} = 0.65$$

Question 8

In a drug trial, 40 people received the non-placebo and 30 of them improved. What is the probability that a person improved given they received the non-placebo?

Accepted Answer

$$P(	ext{Improved} \,|\, 	ext{Non-Placebo}) = \frac{30}{40} = 0.75$$

Question 9

Which of the following best describes a marginal probability in the context of a contingency table?

Accepted Answer

The probability of a single event occurring, regardless of the other variable

Question 10

A survey of 100 students found that 30 are seniors, 40 drive a car, and 15 are seniors who drive a car. What is the probability that a randomly selected student is a senior given that they drive a car?

Accepted Answer

$$P(	ext{Senior} \,|\, 	ext{Drives}) = \frac{15}{40} = 0.375$$

Question 11

In a contingency table, which type of probability is calculated by dividing the frequency of a cell by the total frequency in its row or column?

Accepted Answer

Conditional probability

Question 12

A table shows that 10 people are vegan and gluten-free, 6 are vegan and not gluten-free, 8 are not vegan and gluten-free, and 96 are neither. What is the probability that a randomly selected person is vegan or gluten-free?

Accepted Answer

$$P(	ext{Vegan or Gluten-Free}) = \frac{10+6+8}{120} = \frac{24}{120} = 0.2$$

Question 13

If a contingency table shows that 20 people improved with placebo, 30 did not improve with placebo, 30 improved with non-placebo, and 20 did not improve with non-placebo, what is the joint probability that a person improved and received the non-placebo?

Accepted Answer

$$P(	ext{Improved} \cap 	ext{Non-Placebo}) = \frac{30}{100} = 0.3$$

Question 14

Which of the following statements is true regarding the use of contingency tables in statistics?

Accepted Answer

They are used to display the frequency distribution of variables and to calculate marginal, joint, and conditional probabilities.

Question 15

A survey of 120 guests at a wedding found that 78 are vegetarian, 16 are vegan, and 26 eat meat. If 10 vegetarians are also vegan, what is the probability that a randomly selected guest is either vegetarian or vegan?

Accepted Answer

$$P(	ext{Vegetarian or Vegan}) = \frac{78+16-10}{120} = \frac{84}{120} = 0.7$$

Analyzing Probabilities Using Contingency Tables

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