Q: A deck of cards: Three cards are drawn in succession without replacement. What is the probability that the first card is a red ace, the second is a 10 or jack, and the third is greater than 3 but less than 7?

$$\frac{2}{52} \times \frac{8}{51} \times \frac{12}{50}$$

Q: A medical test for a disease has a 99% sensitivity and 95% specificity. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease? (Use Bayes' Theorem.)

$$\frac{0.99 \times 0.01}{0.99 \times 0.01 + 0.05 \times 0.99}$$

Question 1

Suppose events $$A$$ and $$B$$ are independent, with $$P(A \cap B') = 0.2$$ and $$P(A' \cap B) = 0.3. $$What is $$P(A \cup B)$$?

Accepted Answer

$$0.6$$

Question 2

In a box are 11 balls numbered 1 to 11. Three balls are drawn without replacement. What is the probability that the second ball is odd given that the first ball is even?

Accepted Answer

$$\frac{6}{10}$$

Question 3

A flight has $$P(D) = 0.83$$ for departing on time, $$P(A) = 0.82$$ for arriving on time, and $$P(D \cap A) = 0.78. $$What is the probability the plane arrives on time given it departed on time?

Accepted Answer

$$\frac{0.78}{0.83}$$

Question 4

Which of the following is TRUE about mutually exclusive and independent events?

Accepted Answer

Mutually exclusive events cannot be independent unless one has probability zero.

Question 5

A bag contains 4 white and 3 black balls. A second bag contains 3 white and 5 black balls. One ball is drawn from the first bag and placed into the second bag. What is the probability that a ball now drawn from the second bag is black?

Accepted Answer

$$\frac{33}{63}$$

Question 6

Let $$A$$ and $$B be $$independent events. Which of the following is always true?

Accepted Answer

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

Question 7

Suppose $$P(A|B) = 0.4$$ and $$P(B) = 0.5. $$What is $$P(A \cap B)$$?

Accepted Answer

$$0.2$$

Question 8

Given events $$A$$ and $$B$$ with $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.3$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.8$$

Question 9

Suppose $$P(A|B \cap C) = \frac{P(B|A \cap C)P(A|C)}{P(B|C)}. $$This formula is an example of:

Accepted Answer

Bayes' Theorem

Question 10

A deck of cards: Three cards are drawn in succession without replacement. What is the probability that the first card is a red ace, the second is a 10 or jack, and the third is greater than 3 but less than 7?

Accepted Answer

$$\frac{2}{52} 	imes \frac{8}{51} 	imes \frac{12}{50}$$

Question 11

Suppose $$P(B_1$) = 0.3$$, $$P(B_2$) = 0.7$$, $$P(A|B_1$) = 0.2$$, $$P(A|B_2$) = 0.5. $$What is $$P(A) by $$the Law of Total Probability?

Accepted Answer

$$0.41$$

Question 12

A medical test for a disease has a 99% sensitivity and 95% specificity. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease? (Use Bayes' Theorem.)

Accepted Answer

$$\frac{0.99 	imes 0.01}{0.99 	imes 0.01 + 0.05 	imes 0.99}$$

Question 13

If $$P(A|B) = 0.7$$ and $$P(B) = 0.4$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.28$$

Question 14

Which of the following is the correct statement of the Law of Total Probability?

Accepted Answer

$$P(A) = \sum_i P(A \cap B_i) = \sum_i P(A|B_i)P(B_i)$$

Question 15

A company finds that 80% of its insureds are good drivers. The probability that a bad driver files a claim is $$0.7. $$What is the probability that a randomly selected insured is a bad driver and does not file a claim?

Accepted Answer

$$0.14$$

Question 16

If $$P(A|B) + P(A^c|B) = 1$$, what does this express?

Accepted Answer

The sum of the probabilities of complementary events given $$B is 1.$$

Question 17

If $$P(A \cap B) = P(A)P(B)$$, what can be concluded about $$A$$ and $$B$$?

Accepted Answer

They are independent.

Question 18

A tree diagram is useful in probability because:

Accepted Answer

It helps visualize conditional probabilities and multiply along paths to find joint probabilities.

Question 19

If $$P(A|B) = P(A)$$, what does this imply?

Accepted Answer

Events $$A$$ and $$B$$ are independent.

Question 20

Suppose $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2. $$What is $$P(A|B)$$?

Accepted Answer

$$0.4$$

Guided Practice: Conditional Probability, Independence, and Bayes' Theorem in Actuarial and Health Contexts

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