Q: If $$P(A) = 0.7$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.35$$, are $$A$$ and $$B$$ independent?

Yes, because $$P(A \cap B) = P(A) \times P(B)$$

Question 1

Suppose you roll a fair six-sided die. What is the probability that the number 3 does not show up?

Accepted Answer

$$\frac{5}{6}$$

Question 2

If $$A is $$the event 'the sum of two dice is at least 9', what is $$P(A)$$?

Accepted Answer

Question 3

Let $$A be $$the event 'two faces are the same' when rolling two dice. What is $$P(A)$$?

Accepted Answer

$$\frac{6}{36}$$

Question 4

A box contains 3 blue balls and 2 red balls. If two balls are drawn at random without replacement, what is the probability that both are blue?

Accepted Answer

$$\frac{3}{10}$$

Question 5

If $$A$$ and $$B$$ are independent events with $$P(A) = 0.4$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.2$$

Question 6

Given the following probabilities: $$P(M) = 0.6$$, $$P(F) = 0.4$$, $$P(M \cap S) = 0.3$$, $$P(F \cap S) = 0.2$$, what is $$P(S)$$?

Accepted Answer

$$0.5$$

Question 7

A medical test for a disease has a 90% chance of detecting the disease when it is present (sensitivity) and a 95% chance of correctly identifying a healthy person (specificity). If 1% of the population has the disease, what is the probability that a randomly selected person tests positive?

Accepted Answer

$$0.0585$$

Question 8

If $$P(A) = 0.7$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.35$$, are $$A$$ and $$B$$ independent?

Accepted Answer

Yes, because $$P(A \cap B) = P(A) 	imes P(B)$$

Question 9

A tree diagram is used to represent which of the following?

Accepted Answer

All possible outcomes of a sequence of events and their probabilities

Question 10

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

Accepted Answer

$$0.1$$

Question 11

Suppose a random variable $$X$$ has the following distribution: $$P(X=0) = 0.5$$, $$P(X=1) = 0.3$$, $$P(X=2) = 0.2. $$What is the expected value $$E(X)$$?

Accepted Answer

$$0.7$$

Question 12

A game pays $$12 if you win and costs 2 to $$play. The probability of winning is $$\frac{1}{6}. $$What should the cost to play be for the game to be fair?

Accepted Answer

$$\4$

Question 13

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cup B) = 0.58$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.12$$

Question 14

In a class, 60% of students take Math, 40% take English, and 20% take both. What is the probability that a randomly selected student takes Math or English?

Accepted Answer

$$0.8$$

Question 15

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

Accepted Answer

$$0$$

Question 16

A probability distribution for $$X is $$given by $$P(X=0) = 0.2$$, $$P(X=1) = 0.5$$, $$P(X=2) = 0.3. $$What is the variance of $$X$$?

Accepted Answer

$$0.46$$

Question 17

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

Accepted Answer

$$0.2$$

Question 18

A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?

Accepted Answer

$$\frac{2}{15}$$

Question 19

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12$$, what is $$P(A|B)$$?

Accepted Answer

$$0.3$$

Question 20

A student is either a freshman or a sophomore. 60% are freshmen, and 40% are sophomores. If 70% of freshmen and 50% of sophomores pass a test, what is the probability that a randomly selected student passes?

Accepted Answer

$$0.62$$

Probability and Discrete Random Variables: Review and Applications

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