Q: A plant manager receives a shipment of 100 circuits, 5 of which are defective. She randomly selects 2 circuits without replacement. What is the probability that at least one of the two circuits is defective?

$$1 - \left(\frac{95}{100} \times \frac{94}{99}\right)$$

Q: Which of the following is the correct formula for the conditional probability of $$F$$ given $$E$$?

$$P(F|E) = \frac{P(E \text{ and } F)}{P(E)}$$

Q: If $$E_1$$, $$E_2$$, ..., E_n$$ are independent events, what is the probability that all of them occur?

$$P(E_1$) \times P(E_2$) \times \cdots \times P(E_n)$$

Q: Which of the following is the General Multiplication Rule for any two events $$E$$ and $$F$$?

$$P(E \text{ and } F) = P(E) \times P(F|E)$$

Question 1

Suppose you roll a fair six-sided die twice. Let $$A be $$the event '6 on the first roll' and $$B be $$the event '6 on the second roll.' What is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$\frac{1}{12}$$

Question 2

Which of the following statements is TRUE regarding independent and disjoint (mutually exclusive) events?

Accepted Answer

If two events are disjoint and both have nonzero probability, they cannot be independent.

Question 3

A plant manager receives a shipment of 100 circuits, 5 of which are defective. She randomly selects 2 circuits without replacement. What is the probability that at least one of the two circuits is defective?

Accepted Answer

$$1 - \left(\frac{95}{100} 	imes \frac{94}{99}ight)$$

Question 4

If $$P(E) = 0.4$$ and $$P(F) = 0.7$$, and no further information is given, what is the largest possible value for $$P(E 	ext{ and } F)$$?

Accepted Answer

$$0.4$$

Question 5

Suppose the probability that a randomly selected 24-year-old male survives the year is $$0.9986. $$What is the probability that at least one out of 1000 randomly selected 24-year-old males dies during the year?

Accepted Answer

$$1 - (0.9986$$)^{1000}$$

Question 6

Which of the following is the correct formula for the conditional probability of $$F$$ given $$E$$?

Accepted Answer

$$P(F|E) = \frac{P(E 	ext{ and } F)}{P(E)}$$

Question 7

A driver who is speeding has a probability of $$0.8 of $$being pulled over. If pulled over, the probability of getting a ticket is $$0.9. $$What is the probability that a speeding driver is pulled over and gets a ticket?

Accepted Answer

$$0.8 	imes 0.9 = 0.72$$

Question 8

If $$E_1$$, $$E_2$$, ..., E_n$$ are independent events, what is the probability that all of them occur?

Accepted Answer

$$P(E_1$) 	imes P(E_2$) 	imes \cdots 	imes P(E_n)$$

Question 9

A standard deck of 52 cards is shuffled. You draw one card, replace it, reshuffle, and draw a second card. What is the probability that both cards are hearts?

Accepted Answer

$$\left(\frac{13}{52}\$$right)^2$$

Question 10

Suppose you roll a die and are told the outcome is odd. What is the probability that the outcome is 3?

Accepted Answer

$$\frac{1}{3}$$

Question 11

Which of the following is the General Multiplication Rule for any two events $$E$$ and $$F$$?

Accepted Answer

$$P(E 	ext{ and } F) = P(E) 	imes P(F|E)$$

Question 12

If two events $$E$$ and $$F$$ are independent, which of the following must be true?

Accepted Answer

$$P(E|F) = P(E)$$

Question 13

A sample of size 2 is drawn without replacement from a set labeled 1, 2, 3, 4, 5. Let $$A be$$ '1 is chosen' and $$B be$$ '2 is chosen.' Are $$A$$ and $$B$$ independent?

Accepted Answer

No, because the occurrence of $$A$$ affects the probability of $$B.$$

Question 14

In 2005, 19.1% of all murder victims were between 20 and 24 years old, and 16.6% were 20-24 year old males. What is the probability that a randomly selected murder victim was male, given that the victim was 20-24 years old?

Accepted Answer

$$\frac{0.166}{0.191}$$

Question 15

If the probability of an event $$E is$$ $$0.2$$, what is the probability that $$E$$ does not occur?

Accepted Answer

$$1 - 0.2 = 0.8$$

Question 16

Which of the following best describes the difference between independent and dependent events?

Accepted Answer

For independent events, the occurrence of one does not affect the probability of the other; for dependent events, it does.

Question 17

If you randomly select 2 cards from a deck, replacing and reshuffling after the first draw, what assumption allows you to multiply the probabilities of each draw?

Accepted Answer

The draws are independent because of replacement and reshuffling.

Question 18

If a small random sample is taken from a large population without replacement, under what condition can the events be treated as independent?

Accepted Answer

If the sample size is less than 5% of the population size.

Question 19

Which rule would you use to find the probability that at least one of several independent events occurs?

Accepted Answer

Use the complement rule: $$P(	ext{at least one}) = 1 - P(	ext{none})$$

Question 20

If $$P(E) = 0.5$$ and $$P(F|E) = 0.2$$, what is $$P(E 	ext{ and } F)$$?

Accepted Answer

$$0.5 	imes 0.2 = 0.1$$

Lecture 13: Independence and Conditional Probability

Study Guide - Practice Questions

Study Guide - Flashcards