Q: Which formula correctly expresses the probability of 'at least one' occurrence of event A in n independent trials?

$$P(\text{at least one A}) = 1 - P(\text{no A in n trials})$$

Q: Which of the following is the correct formula for conditional probability?

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

Question 1

Which of the following best describes the complement of the event 'at least one defective product in a batch of 10'?

Accepted Answer

None of the 10 products are defective.

Question 2

A company has a 12% defect rate in its products. If a customer buys 5 products, what is the probability that at least one is defective? Use the complement rule.

Accepted Answer

$$1 - (0.88)^5$

Question 3

Which formula correctly expresses the probability of 'at least one' occurrence of event A in n independent trials?

Accepted Answer

$$P(	ext{at least one A}) = 1 - P(	ext{no A in n trials})$$

Question 4

If the probability that a manufactured item is not defective is 0.97, what is the probability that in a random sample of 8 items, all are not defective?

Accepted Answer

$$(0.97)^8$$

Question 5

A medical test for a disease has a 90% chance of correctly identifying a user who actually has the disease. If 100 people with the disease are tested, what is the expected number of people who will test positive?

Accepted Answer

$$100 	imes 0.90 = 90$$

Question 6

Given $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(B|A)$$?

Accepted Answer

$$\frac{0.2}{0.4} = 0.5$$

Question 7

In a pre-employment drug screening, 45 out of 50 drug users tested positive. What is the probability that a drug user tests positive?

Accepted Answer

$$\frac{45}{50} = 0.9$$

Question 8

If 70 people tested positive for drugs and 45 of them actually use drugs, what is the probability that a person who tested positive actually uses drugs?

Accepted Answer

$$\frac{45}{70} \approx 0.643$$

Question 9

Which of the following statements is TRUE regarding conditional probability?

Accepted Answer

$$P(B|A) is $$generally not equal to $$P(A|B).$$

Question 10

A batch of 20 light bulbs has a 5% defect rate. If a customer buys 4 bulbs, what is the probability that none are defective?

Accepted Answer

$$(0.95)^4$$

Question 11

Which of the following is the correct formula for conditional probability?

Accepted Answer

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

Question 12

A hospital screens for a rare disease. The probability that a randomly selected patient has the disease is 0.01. If 5 patients are tested, what is the probability that at least one has the disease?

Accepted Answer

$$1 - (0.99)^5$

Question 13

Which of the following best describes the intuitive approach to conditional probability?

Accepted Answer

Assume the given condition has occurred and focus only on the relevant subset.

Question 14

A quality control inspector selects 3 items at random from a shipment with a 10% defect rate. What is the probability that at least one item is defective?

Accepted Answer

$$1 - (0.9)^3$

Question 15

If $$P(A) = 0.3$$, $$P(B) = 0.6$$, and $$P(A 	ext{ and } B) = 0.18$$, what is $$P(A|B)$$?

Accepted Answer

$$\frac{0.18}{0.6} = 0.3$$

Probability: Complements, Conditional Probability, and Applications

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