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

$$0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: For a normal distribution with mean $$\mu = 68.4$$ MPH and standard deviation $$\sigma = 7.3$$ MPH, what speed is 1.6 standard deviations above the mean?

$$68.4 + 1.6 \times 7.3 = 80.08$$ MPH

Q: If the probability of making one basket in a free-throw challenge is 0.42, the probability of making two baskets is 0.11, and the winnings are $$50 for one basket and 500 $$for two baskets, what is the expected winnings?

$$0 \times 0.47 + 50 \times 0.42 + 500 \times 0.11 = 0 + 21 + 55 = 76$$

Question 1

A bag contains 3 red balls, 2 green balls, and 1 blue ball. If one ball is drawn at random, what is the probability that it is either red or blue?

Accepted Answer

$$\frac{3}{6} + \frac{1}{6} = \frac{4}{6}$$

Question 2

Which of the following statements best describes the Law of Large Numbers?

Accepted Answer

As the number of trials increases, the proportion of times a particular outcome occurs gets closer to the true probability.

Question 3

If $$P(A) = 0.4$$, what is $$P(A^c)$$, the probability of the complement of event A?

Accepted Answer

$$0.6$$

Question 4

A card is drawn from a standard deck of 52 cards. What is the probability that the card is a face card (Jack, Queen, or King)?

Accepted Answer

$$\frac{12}{52}$$

Question 5

Which of the following pairs of events are mutually exclusive?

Accepted Answer

Drawing a red card and drawing a black card from a deck

Question 6

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.1$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.5 - 0.1 = 0.7$$

Question 7

Suppose the probability that a randomly selected person at a mall is Indian is 0.08, and the probability that an Indian person is wearing a mask is 0.35. What is the probability that a randomly selected person is Indian and wearing a mask?

Accepted Answer

$$0.08 	imes 0.35 = 0.028$$

Question 8

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

Accepted Answer

A and B are independent events.

Question 9

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

Accepted Answer

$$0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 10

The amount of time (X) it takes to drive to UGA is normally distributed with mean $$\mu = 70$$ minutes and standard deviation $$\sigma = 3$$ minutes. What is the probability that a commute is between 67 and 73 minutes?

Accepted Answer

Approximately $$0.818 ($$using the Empirical Rule, about 81.8%)

Question 11

For a normal distribution with mean $$\mu = 68.4$$ MPH and standard deviation $$\sigma = 7.3$$ MPH, what speed is 1.6 standard deviations above the mean?

Accepted Answer

$$68.4 + 1.6 	imes 7.3 = 80.08$$ MPH

Question 12

If the probability of making one basket in a free-throw challenge is 0.42, the probability of making two baskets is 0.11, and the winnings are $$50 for one basket and 500 $$for two baskets, what is the expected winnings?

Accepted Answer

$$0 	imes 0.47 + 50 	imes 0.42 + 500 	imes 0.11 = 0 + 21 + 55 = 76$$

Question 13

A random variable X has the following probability distribution: $$P(X=0)=0.1$$, $$P(X=1)=0.3$$, $$P(X=2)=0.4$$, $$P(X=3)=0.2. $$What is $$P(1 \u003c X \leq 3)$$?

Accepted Answer

$$P(X=2) + P(X=3) = 0.4 + 0.2 = 0.6$$

Question 14

If the probability that a car is speeding (speed \u003e 70 MPH) is 0.39, how many cars out of 1000 would you expect to be speeding?

Accepted Answer

$$0.39 	imes 1000 = 390$$

Question 15

Which of the following is NOT a property of the normal distribution?

Accepted Answer

The area under the curve can be greater than 1.

Question 16

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

Accepted Answer

$$0.5 	imes 0.4 = 0.2$$

Question 17

A random variable X has the following probability distribution: $$P(X=0)=0.1$$, $$P(X=1)=0.2$$, $$P(X=2)=0.3$$, $$P(X=3)=0.4. Is $$this a valid probability distribution?

Accepted Answer

Yes, because the probabilities sum to 1.

Probability and Probability Distributions: Core Concepts for Introductory Statistics

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