Question 1

Suppose in a survey of U.S. residents aged 15 and older, there are 14.9 million widowed and 25.5 million divorced individuals out of a total of 263.6 million. What is the probability that a randomly selected individual is either widowed or divorced?

Accepted Answer

$$\frac{14.9 + 25.5}{263.6} = 0.153$$

Question 2

Given the following counts from a population: 128 million males, 14.9 million widowed, and 3.5 million widowed males, out of a total of 263.6 million people. What is the probability that a randomly selected person is either male or widowed?

Accepted Answer

$$\frac{128 + 14.9 - 3.5}{263.6} = 0.528$$

Question 3

Which of the following best describes the complement of an event $$E in $$probability theory?

Accepted Answer

All outcomes in the sample space that are not in $$E$$

Question 4

If $$P(E) = 0.3$$, what is $$P(E^c)$$, the probability of the complement of $$E$$?

Accepted Answer

$$1 - 0.3 = 0.7$$

Question 5

Two events $$E$$ and $$F$$ are independent if:

Accepted Answer

The occurrence of $$E$$ does not affect the probability of $$F$$

Question 6

Suppose the probability that a 40-year-old woman survives the year is $$0.992. $$What is the probability that at least one of two such women dies during the year?

Accepted Answer

$$1 - (0.992)^2$ = 0.016$$

Question 7

If $$P(E) = 0.4$$ and $$P(F) = 0.5$$, and $$E$$ and $$F$$ are independent, what is $$P(E \cap F)$$?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 8

A study finds that the probability a marriage lasts at least 20 years is $$0.35$$, and the probability a person is married is $$0.52. $$What is the probability that a randomly selected person is married and their marriage has lasted at least 20 years?

Accepted Answer

$$0.35 	imes 0.52 = 0.182$$

Question 9

Given the following table of survey results for age and likelihood to buy a product, what is the probability that a randomly selected respondent is aged 35-44 and is 'more likely' to buy the product? (Suppose there are 329 such respondents out of a total of 1329 in the 35-44 age group.)

Accepted Answer

$$\frac{329}{1329} = 0.2476$$

Question 10

If $$P(E \mid F) = 0.7714$$ and $$P(F) = 0.35$$, what is $$P(E \cap F)$$?

Accepted Answer

$$0.7714 	imes 0.35 = 0.270$$

Question 11

Which formula correctly expresses the general multiplication rule for any two events $$E$$ and $$F$$?

Accepted Answer

$$P(E \cap F) = P(E) \cdot P(F \mid E)$$

Question 12

A discrete random variable is best described as:

Accepted Answer

A variable that can take on a countable number of distinct values

Question 13

If the probability that a randomly selected 60-year-old female survives the year is $$0.992$$, what is the probability that at least one out of 500 such women dies during the year?

Accepted Answer

$$1 - (0.992$$)^{500}$$ \approx 0.981$$

Question 14

Which of the following is the correct formula for the probability of the union of two events $$A$$ and $$B$$?

Accepted Answer

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

Question 15

If $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.3$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.6 + 0.5 - 0.3 = 0.8$$

Question 16

In a contingency table, which of the following is true about the calculation of conditional probability $$P(A \mid B)$$?

Accepted Answer

It is the probability of $$A$$ given that $$B$$ has occurred, calculated as $$P(A \cap B) / P(B)$$

Question 17

Which of the following statements is true about dependent events?

Accepted Answer

The occurrence of one event affects the probability of the other event

Question 18

If $$P(E) = 0.2$$ and $$P(F \mid E) = 0.5$$, what is $$P(E \cap F)$$?

Accepted Answer

$$0.2 	imes 0.5 = 0.1$$

Question 19

A survey finds that 27% of people are likely to buy a product, and 27% of those are aged 35-44. What is the probability that a randomly selected person is both likely to buy and aged 35-44?

Accepted Answer

$$0.27 	imes 0.27 = 0.0729$$

Probability Rules and Counting Techniques in Statistics

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