Q: A company workforce consists of 2,750 females and 1,500 clerical workers out of 5,000 total. If 800 are both female and clerical, are the events 'female' and 'clerical' independent?

No, because $$P(F \cap C) \neq P(F)P(C)$$

Q: An entomologist finds a beetle with a rare pattern. $$98\% of $$rare subspecies and $$5\% of $$common subspecies have the pattern. Rare subspecies are $$0.1\% of $$all beetles. What is the probability the beetle is rare given the pattern?

$$\frac{0.98 \times 0.001}{0.98 \times 0.001 + 0.05 \times 0.999}$$

Question 1

Suppose you roll a fair six-sided die. Let event $$A be$$ 'the outcome is three or less' and event $$B be$$ 'the outcome is even.' What is $$A \cap B$$?

Accepted Answer

$$\{2\}$$

Question 2

Given a sample of $$n$$ observations $$x_1$$, $$x_2$$, \ldots, x_n$$, which formula correctly calculates the sample variance?

Accepted Answer

$$s^2$$ = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x}$$)^2$$

Question 3

A company has 100 loans: 15 had a co-applicant (event $$J$$), 48 had a mortgage ($$M$$), and 10 had both. What is the probability that a randomly selected loan has either a co-applicant or a mortgage (or both)?

Accepted Answer

$$P(J \cup M) = 0.53$$

Question 4

Which of the following is NOT a valid event when rolling a fair six-sided die?

Accepted Answer

$$C = \{3, 5, 7\}$$

Question 5

If $$A$$ and $$B$$ are disjoint events, which property of probability applies to $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B)$$

Question 6

A medical test for COVID-19 returns positive for 95 out of 100 infected people and 90 out of 900 uninfected people. What is the probability that a randomly selected person who tests positive actually has COVID-19?

Accepted Answer

$$\frac{95}{185}$$

Question 7

Which formula expresses the conditional probability of $$A$$ given $$B$$?

Accepted Answer

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

Question 8

A manufacturer finds that the probability a garment is defective is $$0.02$$, and the probability an inspector passes a defective garment is $$0.05. $$What is the probability a garment is defective and shipped?

Accepted Answer

$$0.001$$

Question 9

If $$A$$ and $$B$$ are independent events, which equation must be true?

Accepted Answer

$$P(A \cap B) = P(A)P(B)$$

Question 10

A company workforce consists of 2,750 females and 1,500 clerical workers out of 5,000 total. If 800 are both female and clerical, are the events 'female' and 'clerical' independent?

Accepted Answer

No, because $$P(F \cap C) 
eq P(F)P(C)$$

Question 11

Which of the following is the Law of Total Probability for events $$B_1$$, $$B_2$$, ..., B_n$$ partitioning the sample space?

Accepted Answer

$$P(A) = \sum_{i=1}^n P(A|B_i)P(B_i)$$

Question 12

Tomorrow morning will be either sunny ($$0.6) or $$cloudy ($$0.4). $$Probability of rain in the afternoon is $$0.1 if $$sunny, $$0.5 if $$cloudy. What is the probability it will rain in the afternoon?

Accepted Answer

$$0.26$$

Question 13

Which formula is Bayes' Rule for events $$A$$ and $$B$$ with $$P(B) \u003e 0$$?

Accepted Answer

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

Question 14

An entomologist finds a beetle with a rare pattern. $$98\% of $$rare subspecies and $$5\% of $$common subspecies have the pattern. Rare subspecies are $$0.1\% of $$all beetles. What is the probability the beetle is rare given the pattern?

Accepted Answer

$$\frac{0.98 	imes 0.001}{0.98 	imes 0.001 + 0.05 	imes 0.999}$$

Question 15

Which of the following is always true for any probability function $$P$$ and event $$A$$?

Accepted Answer

$$0 \leq P(A) \leq 1$$

Question 16

If $$A is an $$event, what is the probability of its complement $$A^C$$?

Accepted Answer

$$P(A^C) = 1 - P(A)$$

Question 17

Which set operation represents 'the collection of all objects that are in either set $$A or $$set $$B$$'?

Accepted Answer

$$A \cup B$$

Question 18

Given $$n$$ data points $$x_1$$, $$x_2$$, ..., x_n$$, which property of variance is always true?

Accepted Answer

$$s^2$$ \geq 0$$

4.1 Basic Probability Theory and Descriptive Statistics

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