Q: A bag contains 5 white balls and 3 black balls. Two balls are drawn at random without replacement. What is the probability that both balls are black?

$$\frac{3}{8} \times \frac{2}{7}$$

Question 1

Which of the following best describes the concept of 'sampling space' in probability theory?

Accepted Answer

The set of all possible outcomes of a random experiment.

Question 2

If a die is rolled once, what is the probability of getting an even number?

Accepted Answer

$$\frac{1}{2}$$

Question 3

A box contains 3 red balls, 2 blue balls, and 1 green ball. If one ball is drawn at random, what is the probability that it is not blue?

Accepted Answer

$$\frac{4}{6}$$

Question 4

Which formula represents the probability of the union of two events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Question 5

If two events $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cap B)$$?

Accepted Answer

$$0$$

Question 6

A student randomly picks a card from a standard deck. What is the probability that the card is a heart or a king?

Accepted Answer

$$\frac{14}{52}$$

Question 7

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

Accepted Answer

$$0.15$$

Question 8

A fair coin is tossed three times. What is the probability of getting exactly two heads?

Accepted Answer

$$\frac{3}{8}$$

Question 9

Which of the following is NOT a property of probability?

Accepted Answer

$$P(A) \u003e 1$$

Question 10

If a random variable $$X$$ can take values 1, 2, 3, 4, and 5 with equal probability, what is $$P(X = 3)$$?

Accepted Answer

$$\frac{1}{5}$$

Question 11

A medical test for a disease has a sensitivity of 0.95 and a specificity of 0.90. If the disease prevalence is 0.01, what is the probability that a randomly selected person tests positive?

Accepted Answer

$$0.059$$

Question 12

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.7$$

Question 13

A student is absent from class with probability $$0.1 on $$any given day. What is the probability that the student is present for all five days in a week?

Accepted Answer

$$0.9^5$$

Question 14

If a random variable $$X$$ has the probability mass function $$P(X = x) = \frac{1}{n}$$ for $$x = 1, 2, ..., n$$, what is the expected value $$E[X]$$?

Accepted Answer

$$\frac{n+1}{2}$$

Question 15

A fair die is rolled twice. What is the probability that the sum of the numbers is 7?

Accepted Answer

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

Question 16

Which of the following statements about independent events is correct?

Accepted Answer

If $$A$$ and $$B$$ are independent, then $$P(A \cap B) = P(A) 	imes P(B).$$

Question 17

A bag contains 5 white balls and 3 black balls. Two balls are drawn at random without replacement. What is the probability that both balls are black?

Accepted Answer

$$\frac{3}{8} 	imes \frac{2}{7}$$

Question 18

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

Accepted Answer

$$0.3$$

Question 19

A random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is $$E[X]$$?

Accepted Answer

$$2.1$$

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