Q: Suppose you have two bags: Bag A contains 2 red and 4 blue marbles, and Bag B contains 4 red and 2 blue marbles. You randomly pick a bag and then draw a red marble. What is the probability you picked Bag B? Use Bayes' Theorem.

$$P(\text{Bag B} | \text{red}) = \frac{\frac{4}{6} \cdot \frac{1}{2}}{\frac{2}{6} \cdot \frac{1}{2} + \frac{4}{6} \cdot \frac{1}{2}} = \frac{2}{3}$$

Q: A rare disease affects 1 in 1000 people. A test for the disease is 99% accurate. If a person tests positive, what is the probability they actually have the disease? (Assume $$P(\text{positive} | \text{disease}) = 0.99$$, $$P(\text{positive} | \text{no disease}) = 0.01)$$

$$P(\text{disease} | \text{positive}) = \frac{0.99 \times 0.001}{0.99 \times 0.001 + 0.01 \times 0.999} \approx 0.09$$

Q: If $$P(A) = 0.3$$, $$P(B) = 0.6$$, and $$P(B|A) = 0.5$$, what is $$P(A|B)$$?

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.5 \times 0.3}{0.6} = 0.25$$

Q: A company has two machines producing widgets. Machine 1 produces 60% of the widgets and has a defect rate of 2%. Machine 2 produces 40% of the widgets and has a defect rate of 5%. If a widget is found to be defective, what is the probability it was produced by Machine 2?

$$P(M2|D) = \frac{0.05 \times 0.4}{0.02 \times 0.6 + 0.05 \times 0.4} = \frac{0.02}{0.012 + 0.02} = \frac{0.02}{0.032} = 0.625$$

Q: A test for a certain drug is positive 95% of the time when the drug is present and 10% of the time when the drug is absent. If 2% of the population uses the drug, what is the probability that a randomly selected person who tests positive actually uses the drug?

$$P(\text{user} | +) = \frac{0.95 \times 0.02}{0.95 \times 0.02 + 0.10 \times 0.98} \approx 0.162$$

Q: A person is tested for a rare genetic condition. The condition is present in 0.5% of the population. The test is 98% accurate for those with the condition and 97% accurate for those without. What is the probability that a person who tests positive actually has the condition?

$$P(\text{condition} | +) = \frac{0.98 \times 0.005}{0.98 \times 0.005 + 0.03 \times 0.995} \approx 0.14$$

Question 1

Which of the following best describes Bayes' Theorem?

Accepted Answer

A formula that allows us to update the probability of a hypothesis based on new evidence.

Question 2

Suppose you have two bags: Bag A contains 2 red and 4 blue marbles, and Bag B contains 4 red and 2 blue marbles. You randomly pick a bag and then draw a red marble. What is the probability you picked Bag B? Use Bayes' Theorem.

Accepted Answer

$$P(	ext{Bag B} | 	ext{red}) = \frac{\frac{4}{6} \cdot \frac{1}{2}}{\frac{2}{6} \cdot \frac{1}{2} + \frac{4}{6} \cdot \frac{1}{2}} = \frac{2}{3}$$

Question 3

A medical test for a disease is 99% accurate (i.e., $$P(	ext{positive} | 	ext{disease}) = 0.99$$ and $$P(	ext{negative} | 	ext{no disease}) = 0.99). If 1$$% of the population has the disease, what is the probability that a person who tests positive actually has the disease?

Accepted Answer

Question 4

In a city, 10% of people have the flu. A test for the flu is positive 90% of the time if the person has the flu, and 5% of the time if the person does not have the flu. If a person tests positive, what is the probability they actually have the flu?

Accepted Answer

Question 5

A weather report states that 72% of rainy days result in train delays, and 28% of all days are rainy. If a train is delayed, what is the probability it was raining?

Accepted Answer

Question 6

Which of the following is the correct formula for Bayes' Theorem?

Accepted Answer

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

Question 7

A rare disease affects 1 in 1000 people. A test for the disease is 99% accurate. If a person tests positive, what is the probability they actually have the disease? (Assume $$P(	ext{positive} | 	ext{disease}) = 0.99$$, $$P(	ext{positive} | 	ext{no disease}) = 0.01)$$

Accepted Answer

$$P(	ext{disease} | 	ext{positive}) = \frac{0.99 	imes 0.001}{0.99 	imes 0.001 + 0.01 	imes 0.999} \approx 0.09$$

Question 8

If $$P(A) = 0.3$$, $$P(B) = 0.6$$, and $$P(B|A) = 0.5$$, what is $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.5 	imes 0.3}{0.6} = 0.25$$

Question 9

A company has two machines producing widgets. Machine 1 produces 60% of the widgets and has a defect rate of 2%. Machine 2 produces 40% of the widgets and has a defect rate of 5%. If a widget is found to be defective, what is the probability it was produced by Machine 2?

Accepted Answer

$$P(M2|D) = \frac{0.05 	imes 0.4}{0.02 	imes 0.6 + 0.05 	imes 0.4} = \frac{0.02}{0.012 + 0.02} = \frac{0.02}{0.032} = 0.625$$

Question 10

Which of the following is NOT a requirement for applying Bayes' Theorem?

Accepted Answer

The events must be mutually exclusive.

Question 11

A test for a certain drug is positive 95% of the time when the drug is present and 10% of the time when the drug is absent. If 2% of the population uses the drug, what is the probability that a randomly selected person who tests positive actually uses the drug?

Accepted Answer

$$P(	ext{user} | +) = \frac{0.95 	imes 0.02}{0.95 	imes 0.02 + 0.10 	imes 0.98} \approx 0.162$$

Question 12

In the context of Bayes' Theorem, what does the denominator $$P(B)$$ represent?

Accepted Answer

The total probability of event $$B$$ occurring, considering all possible ways it can happen.

Question 13

A person is tested for a rare genetic condition. The condition is present in 0.5% of the population. The test is 98% accurate for those with the condition and 97% accurate for those without. What is the probability that a person who tests positive actually has the condition?

Accepted Answer

$$P(	ext{condition} | +) = \frac{0.98 	imes 0.005}{0.98 	imes 0.005 + 0.03 	imes 0.995} \approx 0.14$$

Question 14

Which of the following real-world scenarios is best modeled using Bayes' Theorem?

Accepted Answer

Updating the probability that a patient has a disease after receiving a positive test result.

Question 15

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(B|A) = 0.8$$, what is $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{0.8 	imes 0.4}{0.5} = 0.64$$

Question 16

A new screening test for a virus is positive in 99% of infected individuals and 2% of healthy individuals. If 0.5% of the population is infected, what is the probability that a randomly selected person who tests positive is actually infected?

Accepted Answer

$$P(	ext{infected} | +) = \frac{0.99 	imes 0.005}{0.99 	imes 0.005 + 0.02 	imes 0.995} \approx 0.20$$

Question 17

Which of the following statements about conditional probability is TRUE?

Accepted Answer

$$P(A|B) is $$the probability of $$A$$ occurring given that $$B$$ has occurred.

Question 18

A bag contains 3 red and 7 blue marbles. Another bag contains 6 red and 4 blue marbles. You randomly select a bag and draw a red marble. What is the probability you chose the second bag?

Accepted Answer

$$P(	ext{Bag 2} | 	ext{red}) = \frac{0.6 	imes 0.5}{0.3 	imes 0.5 + 0.6 	imes 0.5} = \frac{0.3}{0.15 + 0.3} = \frac{0.3}{0.45} = 0.67$$

Bayes' Theorem and Conditional Probability: Study Notes

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