Q: Which formula correctly expresses the multiplication rule for the probability of two dependent events $$A$$ and $$B$$?

$$P(A \text{ and } B) = P(A) \times P(B|A)$$

Q: A coin is tossed and a die is rolled. What is the probability of tossing a head and then rolling a 6?

$$\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

Q: A standard deck of 52 cards is used. If you draw two cards without replacement, what is the probability that the first is a king and the second is a queen?

$$\frac{4}{52} \times \frac{4}{51}$$

Question 1

Which of the following best describes conditional probability?

Accepted Answer

The probability of an event occurring, given that another event has already occurred.

Question 2

Suppose two cards are drawn in sequence from a standard deck of 52 cards without replacement. What is the probability that the second card is a queen, given that the first card is a king?

Accepted Answer

$$\frac{4}{51}$$

Question 3

Which of the following pairs of events are independent?

Accepted Answer

Tossing a coin and getting a head, and then rolling a six-sided die and obtaining a 6.

Question 4

If $$P(A) = 0.7$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.35$$

Question 5

A medical procedure has a success rate of $$0.95. $$What is the probability that three independent surgeries are all successful?

Accepted Answer

$$(0.95)^3$$

Question 6

A medical procedure has a $$0.95$$ probability of success. What is the probability that none of three independent surgeries are successful?

Accepted Answer

$$(0.05)^3$$

Question 7

A medical procedure has a $$0.95$$ probability of success. What is the probability that at least one of three independent surgeries is successful?

Accepted Answer

$$1 - (0.05)^3$

Question 8

In a survey, 225 adults said they have ridden as a passenger in a self-driving vehicle. Of these, 202 are 18 to 64 years old. What is the probability that a randomly selected adult who has ridden as a passenger in a self-driving vehicle is 18 to 64 years old?

Accepted Answer

$$\frac{202}{225}$$

Question 9

Which formula correctly expresses the multiplication rule for the probability of two dependent events $$A$$ and $$B$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B|A)$$

Question 10

A coin is tossed and a die is rolled. What is the probability of tossing a head and then rolling a 6?

Accepted Answer

$$\frac{1}{2} 	imes \frac{1}{6} = \frac{1}{12}$$

Question 11

Which of the following statements is true about independent events?

Accepted Answer

The occurrence of one event does not affect the probability of the other event.

Question 12

A standard deck of 52 cards is used. If you draw two cards without replacement, what is the probability that the first is a king and the second is a queen?

Accepted Answer

$$\frac{4}{52} 	imes \frac{4}{51}$$

Question 13

A recent year saw 19,326 U.S. MD medical school seniors apply to residency programs. Of these, 18,108 matched with residency positions, and about 75.6% of those matched with one of their top three choices. What is the probability that a randomly selected senior was matched with a residency position and it was one of their top three choices?

Accepted Answer

$$0.708$$

Question 14

If $$P(B|A) = 0.756$$, what is the probability that a senior matched to a residency position did NOT get one of their top three choices?

Accepted Answer

$$1 - 0.756 = 0.244$$

Question 15

Which of the following is an example of dependent events?

Accepted Answer

Drawing two cards from a deck without replacement.

Question 16

If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.2$$

Question 17

A driver is speeding (event A) and then gets into a car accident (event B). Which of the following is true?

Accepted Answer

The events are dependent because speeding increases the chance of an accident.

Question 18

If $$P(A) = 0.6$$, $$P(B|A) = 0.3$$, and $$P(B|A') = 0.2$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.18$$

Question 19

Which of the following is the correct interpretation of $$P(B|A)$$?

Accepted Answer

The probability of event B occurring given that event A has occurred.

Question 20

A student claims that the probability of getting at least one successful surgery out of three independent surgeries, each with a $$0.95$$ success rate, is $$0.95^3. Is $$this correct?

Accepted Answer

No, the correct probability is $$1 - (0.05)^3$.

Conditional Probability and the Multiplication Rule (Elementary Statistics, Chapter 3.2)

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