Q: A medical test for a rare disease has a 99% sensitivity (true positive rate) and a 95% specificity (true negative rate). The disease prevalence is 1 in 1000. If a patient tests positive, what is the probability they actually have the disease? (Use Bayes’ Rule.)

$$\dfrac{0.99 \times 0.001}{0.99 \times 0.001 + 0.05 \times 0.999}$$

Q: In the circuit example, what is the probability that both randomly selected circuits are not defective?

$$\dfrac{95}{100} \times \dfrac{94}{99}$$

Q: A fair coin is flipped. If heads, a ball is drawn from urn 1 (2 red, 4 white); if tails, a ball is drawn from urn 2 (5 red, 1 blue). What is the probability of drawing a blue ball?

$$\dfrac{1}{2} \times \dfrac{1}{6}$$

Q: Suppose you draw a ball and it is red. What is the probability that the coin flip was heads? (Urn 1: 2 red, 4 white; Urn 2: 5 red, 1 blue)

$$\dfrac{\frac{1}{2} \times \frac{2}{6}}{\frac{1}{2} \times \frac{2}{6} + \frac{1}{2} \times \frac{5}{6}}$$

Q: Suppose $$P(E) = 0.3$$, $$P(F) = 0.5$$, and $$P(E \cap F) = 0.15. $$Are $$E$$ and $$F$$ independent?

Yes, because $$P(E \cap F) = P(E) \times P(F)$$

Q: A shipment of 100 items contains 5 defective. If two are randomly selected and tested, what is the probability that at least one is defective?

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

Question 1

Suppose you roll two fair dice. Let event A be 'the first die shows an even number' and event B be 'the sum of the dice is 7'. What is $$P(A \mid B)$$?

Accepted Answer

$$\dfrac{1}{2}$$

Question 2

A medical test for a rare disease has a 99% sensitivity (true positive rate) and a 95% specificity (true negative rate). The disease prevalence is 1 in 1000. If a patient tests positive, what is the probability they actually have the disease? (Use Bayes’ Rule.)

Accepted Answer

$$\dfrac{0.99 	imes 0.001}{0.99 	imes 0.001 + 0.05 	imes 0.999}$$

Question 3

Which of the following correctly states the Law of Total Probability for three mutually exclusive and exhaustive events $$A_1$$, $$A_2$$, $$A_3$$?

Accepted Answer

$$P(E) = P(A_1$) \cdot P(E \mid A_1$) + P(A_2$) \cdot P(E \mid A_2$) + P(A_3$) \cdot P(E \mid A_3$)$$

Question 4

In the circuit example, what is the probability that both randomly selected circuits are not defective?

Accepted Answer

$$\dfrac{95}{100} 	imes \dfrac{94}{99}$$

Question 5

If events $$E$$ and $$F$$ are independent, which of the following is always true?

Accepted Answer

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

Question 6

A fair coin is flipped. If heads, a ball is drawn from urn 1 (2 red, 4 white); if tails, a ball is drawn from urn 2 (5 red, 1 blue). What is the probability of drawing a blue ball?

Accepted Answer

$$\dfrac{1}{2} 	imes \dfrac{1}{6}$$

Question 7

Which of the following is the correct formula for conditional probability $$P(A \mid B)$$?

Accepted Answer

$$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$$

Question 8

Suppose you draw a ball and it is red. What is the probability that the coin flip was heads? (Urn 1: 2 red, 4 white; Urn 2: 5 red, 1 blue)

Accepted Answer

$$\dfrac{\frac{1}{2} 	imes \frac{2}{6}}{\frac{1}{2} 	imes \frac{2}{6} + \frac{1}{2} 	imes \frac{5}{6}}$$

Question 9

Which statement about partitions is correct?

Accepted Answer

Events in a partition are pairwise disjoint and their union is the entire sample space.

Question 10

If you take a random sample of size 20 from a population of 1000 without replacement, can you treat the events as independent?

Accepted Answer

Yes, because $$\frac{20}{1000} = 2\% \u003c 5\% so $$independence is a reasonable assumption.

Question 11

Which of the following is NOT a requirement for events to form a partition of the sample space?

Accepted Answer

Events must be independent.

Question 12

Suppose $$P(E) = 0.3$$, $$P(F) = 0.5$$, and $$P(E \cap F) = 0.15. $$Are $$E$$ and $$F$$ independent?

Accepted Answer

Yes, because $$P(E \cap F) = P(E) 	imes P(F)$$

Question 13

A shipment of 100 items contains 5 defective. If two are randomly selected and tested, what is the probability that at least one is defective?

Accepted Answer

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

Question 14

Which of the following is the correct statement of Bayes’ Rule for events $$A$$ and $$B$$?

Accepted Answer

$$P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B \mid A) P(A) + P(B \mid A^c) P(A^c)}$$

Question 15

If $$P(E \mid F) = P(E)$$, what does this imply about $$E$$ and $$F$$?

Accepted Answer

They are independent events.

Question 16

In the urn example, what is the probability of drawing a white ball?

Accepted Answer

$$\dfrac{1}{2} 	imes \dfrac{4}{6}$$

Question 17

Suppose $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \mid B) = 0.8. $$What is $$P(A \cap B)$$?

Accepted Answer

$$0.8 	imes 0.5 = 0.4$$

Question 18

Which of the following is a correct application of the general multiplication rule?

Accepted Answer

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

Conditional Probability, Independence, Law of Total Probability, and Bayes’ Rule

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