Q: Given events $$A$$ and $$B$$, which formula correctly expresses the conditional probability of $$A$$ given $$B$$?

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

Q: A diagnostic blood test for Down syndrome yields the following probabilities: $$P(POS \text{ and } D) = 0.009$$, $$P(D) = 0.010$$, and $$P(POS) = 0.257. $$Are the events POS (positive test) and D (Down syndrome present) independent?

No, because $$P(POS|D) \neq P(POS)$$

Q: Which formula is used to find the probability that both events $$A$$ and $$B$$ occur?

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

Question 1

Suppose you roll a fair six-sided die. What is the probability of rolling a 6, given that the outcome is an even number?

Accepted Answer

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

Question 2

In a sample of 40 vehicles, 15 are red, 6 are trucks, and 2 are both red and trucks. If a randomly selected vehicle is red, what is the probability that it is a truck?

Accepted Answer

$$\frac{2}{15}$$

Question 3

Given events $$A$$ and $$B$$, which formula correctly expresses the conditional probability of $$A$$ given $$B$$?

Accepted Answer

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

Question 4

A diagnostic blood test for Down syndrome yields the following probabilities: $$P(POS 	ext{ and } D) = 0.009$$, $$P(D) = 0.010$$, and $$P(POS) = 0.257. $$Are the events POS (positive test) and D (Down syndrome present) independent?

Accepted Answer

No, because $$P(POS|D) 
eq P(POS)$$

Question 5

In the tax audit example, what is the probability that a taxpayer with income at least $$1,000,000 is $$audited? Use $$P(A 	ext{ and } B) = 0.00027$$ and $$P(B) = 0.00249.$$

Accepted Answer

$$0.1084$$

Question 6

Which of the following statements is true about independent events $$A$$ and $$B$$?

Accepted Answer

$$P(A|B) = P(A)$$

Question 7

Novak Djokovic faults on his first serve $$38\% of $$the time. Given that he faults on the first serve, he faults on the second serve $$4.5\% of $$the time. What is the probability he makes a double fault?

Accepted Answer

$$0.017$$

Question 8

Which formula is used to find the probability that both events $$A$$ and $$B$$ occur?

Accepted Answer

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

Question 9

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

Accepted Answer

They are independent

Question 10

A blood test for Down syndrome in pregnant women aged 35 or older yields $$1355$$ positive results out of $$5282$$ tested. What is the probability that a randomly selected woman tests positive?

Accepted Answer

$$0.257$$

Question 11

Which of the following best describes sampling without replacement?

Accepted Answer

Once a subject is selected, it cannot be selected again

Question 12

If $$P(A 	ext{ and } B) = P(A) 	imes P(B)$$, what does this imply about events $$A$$ and $$B$$?

Accepted Answer

They are independent

Question 13

In a contingency table, the cell probabilities are calculated by dividing the cell frequency by the total number of observations. If a cell has frequency $$1,233$$ and the total is $$145,781$$, what is the cell probability?

Accepted Answer

$$0.008$$

Question 14

Which of the following is NOT a correct way to check for independence between events $$A$$ and $$B$$?

Accepted Answer

$$P(A|B) = P(B)$$

Question 15

A false positive in a medical test occurs when:

Accepted Answer

The test states the condition is present, but it is actually absent

Question 16

If $$P(A|B) 
eq P(A)$$, what does this indicate about events $$A$$ and $$B$$?

Accepted Answer

They are dependent

Question 17

In the context of conditional probability, what does the vertical slash '|' represent in $$P(A|B)$$?

Accepted Answer

Given

Question 18

A tennis player faults on the first serve $$38\% of $$the time and, given a first fault, faults on the second serve $$4.5\% of $$the time. What is the probability of a double fault using the multiplication rule?

Accepted Answer

$$0.38 	imes 0.045 = 0.017$$

Question 19

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

Accepted Answer

The probability of event $$A$$ occurring, given that event $$B$$ has occurred

(Lecture 11) Conditional Probability and Independence

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