Q: In a disease testing scenario, which formula correctly calculates the false negative rate?

$$P(\text{negative test} | \text{disease present})$$

Q: Which of the following is the correct formula for the probability of 'at least one' event occurring?

$$P(\text{at least one}) = 1 - P(\text{none})$$

Q: A factory produces phones: Factory A makes 60% of phones with a 2% defect rate, Factory B makes 40% with a 5% defect rate. If a phone is found defective, what is the probability it came from Factory A?

$$\frac{0.6 \times 0.02}{0.6 \times 0.02 + 0.4 \times 0.05}$$

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

Given the formula for conditional probability $$P(B|A) = \frac{P(A \cap B)}{P(A)}$$, what does $$P(A \cap B)$$ represent?

Accepted Answer

The probability that both events A and B occur.

Question 3

A polygraph test is used to detect lies. If $$P(	ext{positive test} | 	ext{truthful}) is $$calculated, what does this probability represent?

Accepted Answer

The probability of a false positive.

Question 4

In a disease testing scenario, which formula correctly calculates the false negative rate?

Accepted Answer

$$P(	ext{negative test} | 	ext{disease present})$$

Question 5

A student is given four quarters. Out of 42 students, 25 spent the money. What is the probability that a student spent the money, given they were given four quarters?

Accepted Answer

$$\frac{25}{42}$$

Question 6

Which of the following is the correct formula for the probability of 'at least one' event occurring?

Accepted Answer

$$P(	ext{at least one}) = 1 - P(	ext{none})$$

Question 7

If the probability of a girl being born is $$0.462$$, what is the probability of at least one boy in 9 births?

Accepted Answer

$$1 - (0.462)^9$

Question 8

A hard disk drive has an 8% failure rate per year. What is the probability that at least one of two independent drives works for a year?

Accepted Answer

$$1 - (0.08)^2$

Question 9

Which statement about independent events is correct?

Accepted Answer

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

Question 10

A combined sample test is performed on three samples, each with a 10% chance of testing positive. What is the probability that the combined test is positive?

Accepted Answer

$$1 - (0.9)^3$

Question 11

Bayes’ Theorem allows us to calculate:

Accepted Answer

The probability of a cause given an observed result.

Question 12

Given Bayes’ Theorem $$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|\overline{A}) \cdot P(\overline{A})}$$, what does the denominator represent?

Accepted Answer

The total probability of observing event B, summed across all possible causes.

Question 13

A factory produces phones: Factory A makes 60% of phones with a 2% defect rate, Factory B makes 40% with a 5% defect rate. If a phone is found defective, what is the probability it came from Factory A?

Accepted Answer

$$\frac{0.6 	imes 0.02}{0.6 	imes 0.02 + 0.4 	imes 0.05}$$

Question 14

In the 'Two Bags of Balls' example, Bag A has 3 red and 7 blue, Bag B has 6 red and 4 blue. If a red ball is drawn, what is the probability it came from Bag A?

Accepted Answer

$$\frac{3/20}{9/20} = \frac{1}{3}$$

Question 15

Which of the following is a common mistake when calculating conditional probability from a table?

Accepted Answer

Using the entire table instead of focusing on the given condition.

Question 16

A phone number’s last four digits are random. What is the probability that at least one digit is 0?

Accepted Answer

$$1 - \left(\frac{9}{10}\$$right)^4$$

Question 17

Which scenario best illustrates the concept of a false negative?

Accepted Answer

A sick patient receives a negative test result and is told they are healthy.

Question 18

If $$P(A) = 0.5$$ and $$P(B) = 0.3$$, and A and B are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.5 	imes 0.3 = 0.15$$

Question 19

Which of the following is NOT a correct application of the complement rule?

Accepted Answer

Calculating the probability of exactly one event occurring.

Conditional Probability, Complements, and Bayes’ Theorem: Study Guide

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