Q: For the lottery in the previous question, if you pay $$2$$ for a ticket, win nothing for 0 matches, win a free ticket for 1 match, $$10$$ for 2 matches, and $$1000$$ for 3 matches, what is your expected profit per ticket?

$$E(\text{profit}) \approx -1.62$$

Question 1

Which of the following best describes a random variable?

Accepted Answer

A variable that assigns a unique numerical value to each outcome of a random experiment.

Question 2

Given the probability distribution for the number of heads in three tosses of a fair coin: $$x = 0, 1, 2, 3$$ with $$P(x=0) = 0.125$$, $$P(x=1) = 0.375$$, $$P(x=2) = 0.375$$, $$P(x=3) = 0.125$$, what is the expected value (mean) of $$x$$?

Accepted Answer

$$E(x) = 1.5$$

Question 3

Which of the following is NOT a requirement for a valid probability distribution of a discrete random variable $$x$$?

Accepted Answer

All probabilities must be equal.

Question 4

A friend bets you $10 that a fair die will land on a 5 or 6. What is the probability that you win the bet (i.e., the die does NOT land on 5 or 6)?

Accepted Answer

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

Question 5

In the Lotto Max lottery, the probability of matching exactly 2 numbers out of 7 is missing from the distribution. Given the probabilities for 0, 1, 3, 4, 5, 6, and 7 matches are $$0.31406$$, $$0.42748$$, $$0.045606$$, $$0.00468$$, $$0.00021$$, $$0.000003$$, and $$0.000000012$$ respectively, what is $$P(x=2)$$?

Accepted Answer

$$0.208641$$

Question 6

Suppose you repeatedly buy one Lotto Max ticket per week. What is the expected number of matches you will observe per ticket, given the probability distribution in the study material?

Accepted Answer

$$E(x) \approx 0.99$$

Question 7

Which of the following random experiments produces a continuous sample space?

Accepted Answer

Measuring the time it takes to commute from home to university.

Question 8

Given the following lottery: 3 tickets are chosen at random from 25, and the probability of matching 0, 1, or 2 numbers is $$0.6697$$, $$0.3013$$, and $$0.0287$$ respectively. What is the probability of matching all 3 numbers?

Accepted Answer

$$0.0003$$

Question 9

For the lottery in the previous question, if you pay $$2$$ for a ticket, win nothing for 0 matches, win a free ticket for 1 match, $$10$$ for 2 matches, and $$1000$$ for 3 matches, what is your expected profit per ticket?

Accepted Answer

$$E(	ext{profit}) \approx -1.62$$

Question 10

Which of the following statements about the probability distribution of a fair die (as shown in the first image) is correct?

Accepted Answer

Each outcome has a probability of $$\frac{1}{6}.$$

Question 11

If $$X is a $$discrete random variable with probability distribution $$P(X = x_i)$$, which formula gives the expected value of $$X$$?

Accepted Answer

$$E(X) = \sum_{i} x_i P(X = x_i)$$

Question 12

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of heads in 10 coin tosses.

Question 13

Suppose 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 $$P(X \geq 1)$$?

Accepted Answer

$$0.8$$

Question 14

If a random variable $$X$$ can take values $$0, 1, 2, 3$$ with probabilities $$0.1, 0.2, 0.3, 0.4$$ respectively, what is the variance of $$X$$?

Accepted Answer

$$	ext{Var}(X) = 1.1$$

Question 15

Which of the following best describes the shape of the probability distribution for the number of matches in the Lotto Max lottery (as shown in the second image)?

Accepted Answer

It is right-skewed, with most probability mass at 0 and 1 matches.

Probability Distributions of Discrete Random Variables

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