Q: In a dataset, $$P(Male) = 0.45$$, $$P(English) = 0.55$$, and $$P(Male \cap English) = 0.25. $$Are the events 'Male' and 'English' independent?

No, because $$P(Male \cap English) \neq P(Male) \times P(English).$$

Question 1

Which of the following best describes a joint probability?

Accepted Answer

The probability of two or more events occurring simultaneously.

Question 2

Given that the probability a person is a student is $$0.5$$ and the probability a person is a soccer player is $$\frac{1}{5.5}$$, what is the marginal probability that a randomly selected person is a student?

Accepted Answer

$$0.5$$

Question 3

Which formula correctly represents the conditional probability of event $$A$$ given event $$B$$?

Accepted Answer

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

Question 4

A deck contains 26 red cards. If there are two 10s among the red cards, what is the probability that a red card drawn is a 10?

Accepted Answer

$$\frac{2}{26}$$

Question 5

Which of the following statements about marginal probability is TRUE?

Accepted Answer

Marginal probability is the probability of a single event, ignoring other events.

Question 6

If $$P(A \cap B) = 0.2$$ and $$P(B) = 0.5$$, what is $$P(A|B)$$?

Accepted Answer

$$0.4$$

Question 7

Which scenario best illustrates a conditional probability?

Accepted Answer

The probability that a person is a student given that they are a soccer player.

Question 8

Suppose in a survey, $$P(English \cap Male) = 0.15$$ and $$P(Male) = 0.4. $$What is the probability that a randomly selected male speaks English?

Accepted Answer

$$0.375$$

Question 9

Which of the following is NOT a use of joint, marginal, and conditional probabilities in statistics?

Accepted Answer

To calculate the mean and median of a dataset.

Question 10

If $$P(A) = 0.3$$ and $$P(A|B) = 0.3$$, what can you infer about the relationship between events $$A$$ and $$B$$?

Accepted Answer

Events $$A$$ and $$B$$ are independent.

Question 11

A medical test is positive for 8% of patients. If 2% of patients are both positive and have the disease, what is the probability that a patient has the disease given a positive test?

Accepted Answer

$$0.25$$

Question 12

Which of the following best describes the relationship between conditional probability and joint probability?

Accepted Answer

Conditional probability is the joint probability divided by the marginal probability of the condition.

Question 13

If $$P(Stats \cap Female) = 0.12$$ and $$P(Female) = 0.3$$, what is $$P(Stats|Female)$$?

Accepted Answer

$$0.4$$

Question 14

Which of the following is an example of a marginal probability?

Accepted Answer

The probability that a person is a student, regardless of other characteristics.

Question 15

In a dataset, $$P(Male) = 0.45$$, $$P(English) = 0.55$$, and $$P(Male \cap English) = 0.25. $$Are the events 'Male' and 'English' independent?

Accepted Answer

No, because $$P(Male \cap English) 
eq P(Male) 	imes P(English).$$

Question 16

Which probability would you use to determine the likelihood that a randomly selected person is a soccer player and a student?

Accepted Answer

Joint probability

Question 17

If $$P(A \cap B) = 0.05$$, $$P(B) = 0.2$$, and $$P(A) = 0.3$$, what is $$P(A|B)$$ and are $$A$$ and $$B$$ independent?

Accepted Answer

$$P(A|B) = 0.25$$, and $$A$$ and $$B$$ are not independent.

Question 18

A company wants to know the probability that an employee is both a manager and works in the IT department. Which type of probability should they calculate?

Accepted Answer

Joint probability

Question 19

Which of the following is a real-world example of conditional probability?

Accepted Answer

The probability that a car is electric given that it is red.

Question 20

If $$P(Female) = 0.6$$, $$P(Stats \cap Female) = 0.18$$, what is the probability that a randomly selected female studies statistics?

Accepted Answer

$$0.3$$

Joint, Marginal, and Conditional Probability: Foundations and Applications

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