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

$$\frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} \approx 0.006$$

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: In a recent year, 18,108 out of 19,326 U.S. MD medical school seniors were matched with residency positions. Of those matched, 75.6% got 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?

$$\frac{18,108}{19,326} \times 0.756 \approx 0.708$$

Q: A deck of 52 cards is used. Two cards are drawn without replacement. What is the probability that both cards are hearts?

$$\frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} \approx 0.059$$

Question 1

Which of the following best describes a 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 first card is a king and the second card is a queen?

Accepted Answer

$$\frac{4}{52} 	imes \frac{4}{51} = \frac{16}{2652} \approx 0.006$$

Question 3

In a survey, 225 adults reported having ridden as a passenger in a self-driving vehicle, and 202 of them were aged 18–64. What is the probability that a randomly selected adult who has ridden as a passenger in a self-driving vehicle is 18–64 years old?

Accepted Answer

$$\frac{202}{225} \approx 0.898$$

Question 4

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 5

If $$P(B|A) = P(B)$$, what can you conclude about events A and B?

Accepted Answer

A and B are independent events.

Question 6

A medical study finds that the probability of a successful ACL surgery is 0.95. What is the probability that all three surgeries (on three different patients) are successful, assuming independence?

Accepted Answer

$$0.95^3 = 0.857$

Question 7

A salmon has a probability of 0.85 of swimming successfully through a dam. What is the probability that two salmon both swim successfully through the dam?

Accepted Answer

$$0.85 	imes 0.85 = 0.7225$$

Question 8

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 9

In a recent year, 18,108 out of 19,326 U.S. MD medical school seniors were matched with residency positions. Of those matched, 75.6% got 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

$$\frac{18,108}{19,326} 	imes 0.756 \approx 0.708$$

Question 10

If the probability that a rotator cuff surgery is successful is 0.9, what is the probability that none of three such surgeries are successful?

Accepted Answer

$$(1 - 0.9)^3$ = 0.001$$

Question 11

A deck of 52 cards is used. Two cards are drawn without replacement. What is the probability that both cards are hearts?

Accepted Answer

$$\frac{13}{52} 	imes \frac{12}{51} = \frac{156}{2652} \approx 0.059$$

Question 12

Which of the following is the correct way to compute the probability that at least one of three independent events occurs, given the probability of each event is $$p$$?

Accepted Answer

$$1 - (1 - p)^3$

Question 13

A researcher wants to know if two events, A and B, are independent. She finds $$P(B) = 0.4$$ and $$P(B|A) = 0.25. $$What can she conclude?

Accepted Answer

A and B are dependent events.

Question 14

Which of the following real-world scenarios best illustrates dependent events?

Accepted Answer

Selecting a king from a deck, not replacing it, and then selecting a queen.

Question 15

If the probability of a surgery failing is 0.05, what is the probability that at least one out of three independent surgeries is successful?

Accepted Answer

$$1 - (1 - 0.05)^3$ = 1 - 0.857375 = 0.142625$$

Question 16

A survey finds that 23 out of 745 adults who have NOT ridden as a passenger in a self-driving vehicle are 65 years or older. What is the probability that a randomly selected adult who has not ridden as a passenger in a self-driving vehicle is 65 or older?

Accepted Answer

$$\frac{23}{745} \approx 0.031$$

Question 17

Which of the following statements is TRUE regarding the probability of two independent events A and B both occurring?

Accepted Answer

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

Question 18

Which of the following events would be considered unusual, based on the typical threshold for probability?

Accepted Answer

Selecting a king and then a queen (without replacement) from a standard deck of cards.

Conditional Probability and the Multiplication Rule

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