Q: A factory has two machines, A and B. Machine A produces 60% of products with a 2% defect rate. Machine B produces 40% of products with a 5% defect rate. What is the overall probability that a randomly selected product is defective?

$$0.6 \times 0.02 + 0.4 \times 0.05 = 0.032$$

Q: Historical data shows 20% of customers purchase product B. After an advertising campaign, 40% of customers have seen the campaign. Of those who bought product B, 60% saw the advertisement. What is the probability a customer buys product B given they saw the advertisement?

$$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)} = \frac{0.6 \times 0.2}{0.4} = 0.3$$

Question 1

Which of the following best describes conditional probability?

Accepted Answer

The probability that one event happens given that another event has already happened.

Question 2

Given two events A and B, which formula correctly represents the conditional probability of A given B?

Accepted Answer

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

Question 3

A factory has two machines, A and B. Machine A produces 60% of products with a 2% defect rate. Machine B produces 40% of products with a 5% defect rate. What is the overall probability that a randomly selected product is defective?

Accepted Answer

$$0.6 	imes 0.02 + 0.4 	imes 0.05 = 0.032$$

Question 4

If two events A and B are independent, which of the following is true?

Accepted Answer

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

Question 5

In the Sally Clarke case, the prosecution multiplied the probability of two independent SIDS events. What statistical error did this represent?

Accepted Answer

The prosecutor's fallacy: confusing the probability of two independent events with the probability of innocence.

Question 6

A bank estimates 6% of its customers will default on credit card debt. If the bank takes on 3 new customers, what is the probability that all three will default?

Accepted Answer

$$0.06^3 = 0.000216$

Question 7

A probability tree is used to visualize which type of probability?

Accepted Answer

Conditional probability

Question 8

Historical data shows 20% of customers purchase product B. After an advertising campaign, 40% of customers have seen the campaign. Of those who bought product B, 60% saw the advertisement. What is the probability a customer buys product B given they saw the advertisement?

Accepted Answer

$$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)} = \frac{0.6 	imes 0.2}{0.4} = 0.3$$

Question 9

Which formula represents Bayes’ Theorem for mutually exclusive events $$A_1$$, $$A_2$$, ..., A_k$$ partitioning the sample space and event B?

Accepted Answer

$$P(A_i|B) = \frac{P(B|A_i) \cdot P(A_i)}{P(B)}$$

Question 10

In a credit default analysis, what is the probability of default given a previous late payment? (Use: 347 defaults with late payment, 833 total with late payment)

Accepted Answer

$$\frac{347}{833} \approx 0.417$$

Question 11

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

Accepted Answer

$$0.3 	imes 0.5 = 0.15$$

Question 12

Two dice are thrown, one red and one black. What is the probability that the sum is at least 10?

Accepted Answer

$$\frac{6}{36} = 0.167$$

Question 13

Given that the black die shows 6, what is the probability that the sum is at least 10?

Accepted Answer

$$\frac{3}{6} = 0.5$$

Question 14

Which of the following statements is true regarding $$P(A|B)$$ and $$P(B|A)$$?

Accepted Answer

$$P(A|B) is $$not necessarily equal to $$P(B|A)$$

Question 15

A customer is found to have defaulted. What is the probability that they had a previous late payment? (Use: 347 defaults with late payment, 655 defaults without late payment, 1002 total defaults)

Accepted Answer

$$\frac{347}{1002} \approx 0.346$$

Question 16

If $$P(A|B) = 0.2$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.2 	imes 0.5 = 0.1$$

Question 17

Which scenario best illustrates the use of Bayes’ Theorem?

Accepted Answer

Updating the probability of a disease given a positive test result.

Question 18

In the OJ Simpson trial, the statistic 'only one in 2500 abusive husbands eventually murder their wives' was cited. What is the main issue with using this statistic to argue innocence?

Accepted Answer

It ignores the conditional probability of guilt given the evidence.

Conditional Probability, Independence, and Bayes’ Theorem: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards