Q: Suppose you flip a fair coin. If it lands on heads, you draw a ball from Urn 1 (which contains 2 red and 4 white balls). If it lands on tails, you draw from Urn 2 (which contains 5 red and 1 blue ball). What is the probability of drawing a red ball?

$$\frac{1}{2} \times \frac{2}{6} + \frac{1}{2} \times \frac{5}{6} = \frac{1}{6} + \frac{5}{12} = \frac{7}{12}$$

Q: A medical test for a rare disease has the following properties: 1 out of 20 people has the disease, the test is positive 4 out of 5 times if the person has the disease, and is positive 1 out of 10 times if the person does not have the disease. If a person tests positive, what is the probability they actually have the disease?

$$\frac{\frac{1}{20} \times \frac{4}{5}}{\frac{1}{20} \times \frac{4}{5} + \frac{19}{20} \times \frac{1}{10}} = \frac{0.04}{0.04 + 0.095} = \frac{0.04}{0.135} \approx 0.296$$

Q: In the weather example, days are Sunny (40%), Cloudy (30%), or Rainy (30%). The probability you are late is 10% on Sunny, 20% on Cloudy, and 50% on Rainy days. What is the probability a student is late on a randomly chosen day?

$$0.4 \times 0.1 + 0.3 \times 0.2 + 0.3 \times 0.5 = 0.04 + 0.06 + 0.15 = 0.25$$

Q: Suppose you know a student was late. Using the weather example (Sunny 40%, Cloudy 30%, Rainy 30%; late probabilities 10%, 20%, 50% respectively), what is the probability it was a Rainy day?

$$\frac{0.3 \times 0.5}{0.4 \times 0.1 + 0.3 \times 0.2 + 0.3 \times 0.5} = \frac{0.15}{0.25} = 0.6$$

Q: In the urn problem, what is the probability that the coin landed on heads given that a red ball was drawn?

$$P(H|R) = \frac{P(R|H)P(H)}{P(R)} = \frac{\frac{2}{6} \times \frac{1}{2}}{\frac{7}{12}} = \frac{1/6}{7/12} = \frac{2}{7}$$

Question 1

Suppose you flip a fair coin. If it lands on heads, you draw a ball from Urn 1 (which contains 2 red and 4 white balls). If it lands on tails, you draw from Urn 2 (which contains 5 red and 1 blue ball). What is the probability of drawing a red ball?

Accepted Answer

$$\frac{1}{2} 	imes \frac{2}{6} + \frac{1}{2} 	imes \frac{5}{6} = \frac{1}{6} + \frac{5}{12} = \frac{7}{12}$$

Question 2

Which of the following is the correct statement of the Law of Total Probability for three mutually exclusive and exhaustive events $$A_1$$, $$A_2$$, $$A_3$$ partitioning the sample space $$S$$?

Accepted Answer

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

Question 3

A medical test for a rare disease has the following properties: 1 out of 20 people has the disease, the test is positive 4 out of 5 times if the person has the disease, and is positive 1 out of 10 times if the person does not have the disease. If a person tests positive, what is the probability they actually have the disease?

Accepted Answer

$$\frac{\frac{1}{20} 	imes \frac{4}{5}}{\frac{1}{20} 	imes \frac{4}{5} + \frac{19}{20} 	imes \frac{1}{10}} = \frac{0.04}{0.04 + 0.095} = \frac{0.04}{0.135} \approx 0.296$$

Question 4

Given events $$A_1$$, $$A_2$$, \ldots, A_k$$ form a partition of the sample space S$, which of the following must be true?

Accepted Answer

The events are pairwise disjoint and their union is $$S.$$

Question 5

In the weather example, days are Sunny (40%), Cloudy (30%), or Rainy (30%). The probability you are late is 10% on Sunny, 20% on Cloudy, and 50% on Rainy days. What is the probability a student is late on a randomly chosen day?

Accepted Answer

$$0.4 	imes 0.1 + 0.3 	imes 0.2 + 0.3 	imes 0.5 = 0.04 + 0.06 + 0.15 = 0.25$$

Question 6

Suppose you know a student was late. Using the weather example (Sunny 40%, Cloudy 30%, Rainy 30%; late probabilities 10%, 20%, 50% respectively), what is the probability it was a Rainy day?

Accepted Answer

$$\frac{0.3 	imes 0.5}{0.4 	imes 0.1 + 0.3 	imes 0.2 + 0.3 	imes 0.5} = \frac{0.15}{0.25} = 0.6$$

Question 7

Which of the following is the correct formula for Bayes’ Rule for two events $$A$$ and $$B$$ with $$P(B) \u003e 0$$?

Accepted Answer

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

Question 8

In the urn problem, what is the probability that the coin landed on heads given that a red ball was drawn?

Accepted Answer

$$P(H|R) = \frac{P(R|H)P(H)}{P(R)} = \frac{\frac{2}{6} 	imes \frac{1}{2}}{\frac{7}{12}} = \frac{1/6}{7/12} = \frac{2}{7}$$

Question 9

Which of the following best describes a probability tree diagram?

Accepted Answer

A graphical representation that shows all possible outcomes of a multi-stage experiment and the probabilities associated with each branch.

Question 10

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

Accepted Answer

$$P(A \cap B) = P(A) \cdot P(B)$$

Question 11

A test for a disease has a sensitivity of 80% and a specificity of 90%. The disease prevalence is 5%. If a person tests positive, what is the probability they have the disease? (Round to two decimal places.)

Accepted Answer

$$\frac{0.05 	imes 0.8}{0.05 	imes 0.8 + 0.95 	imes 0.1} = \frac{0.04}{0.04 + 0.095} = \frac{0.04}{0.135} \approx 0.30$$

Question 12

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

Accepted Answer

The events must be independent.

Question 13

Suppose $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12. $$What is $$P(A|B)$$?

Accepted Answer

$$\frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.4} = 0.3$$

Question 14

If $$P(E|A_1$) = 0.2$$, $$P(E|A_2$) = 0.5$$, $$P(A_1$) = 0.7$$, and $$P(A_2$) = 0.3$$, what is $$P(E)$$ using the Law of Total Probability?

Accepted Answer

$$0.7 	imes 0.2 + 0.3 	imes 0.5 = 0.14 + 0.15 = 0.29$$

Question 15

A company has three factories: A (50% of products), B (30%), and C (20%). The probability a product is defective is 2% for A, 3% for B, and 5% for C. If a product is found defective, what is the probability it came from factory C?

Accepted Answer

$$\frac{0.2 	imes 0.05}{0.5 	imes 0.02 + 0.3 	imes 0.03 + 0.2 	imes 0.05} = \frac{0.01}{0.01 + 0.009 + 0.01} = \frac{0.01}{0.029} \approx 0.34$$

Question 16

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

Accepted Answer

The probability that event $$A$$ occurs given that event $$B$$ has occurred.

Question 17

If $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.6 	imes 0.5 = 0.3$$

Question 18

A bag contains 3 red and 2 blue balls. You draw one ball at random. Let $$A be $$the event 'red ball is drawn' and $$B be $$the event 'blue ball is drawn'. Are $$A$$ and $$B$$ independent?

Accepted Answer

No, because $$A$$ and $$B$$ are mutually exclusive.

Law of Total Probability and Bayes’ Rule: Probability, Partitions, and Applications

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