Q: Given the probability distribution below, what is the expected value $$E(X)$$? $$\begin{array}{cc} X & P(X) \\ 2 & 0.45 \\ 4 & 0.35 \\ 6 & 0.15 \\ 8 & 0.05 \\ \end{array}$$

$$E(X) = 2 \times 0.45 + 4 \times 0.35 + 6 \times 0.15 + 8 \times 0.05 = 3.60$$

Q: A game has the following payoffs and probabilities: $$\begin{array}{cc} Payoff & Probability \\ -10 & 0.2 \\ 0 & 0.5 \\ 20 & 0.3 \\ \end{array} Is $$this game fair?

No, because $$E(X) = (-10 \times 0.2) + (0 \times 0.5) + (20 \times 0.3) = -2 + 0 + 6 = 4.$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values are countable and have gaps between them, such as the number of broken eggs in a carton.

Question 2

Suppose a quality control manager inspects 750 items and finds 32 defectives. What is the empirical probability of finding a defective item?

Accepted Answer

$$\frac{32}{750} = 0.043$$

Question 3

If the probability of an event A is $$P(A) = 0.3$$, what is the probability of its complement $$P(\overline{A})$$?

Accepted Answer

$$P(\overline{A}) = 1 - P(A) = 0.7$$

Question 4

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

Accepted Answer

The sum of all probabilities must be greater than 1.

Question 5

A coin is flipped 100 times and lands heads 52 times. What is the empirical probability of getting heads?

Accepted Answer

$$\frac{52}{100} = 0.52$$

Question 6

Which scenario best illustrates the Law of Large Numbers?

Accepted Answer

As you flip a fair coin thousands of times, the proportion of heads approaches 0.5.

Question 7

Given the probability distribution below, what is the expected value $$E(X)$$?

$$\begin{array}{cc}
X & P(X) \
2 & 0.45 \
4 & 0.35 \
6 & 0.15 \
8 & 0.05 \
\end{array}$$

Accepted Answer

$$E(X) = 2 	imes 0.45 + 4 	imes 0.35 + 6 	imes 0.15 + 8 	imes 0.05 = 3.60$$

Question 8

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

Accepted Answer

The time you spend waiting in line at the grocery store.

Question 9

If a probability distribution for the number of exercise machines sold per day at a store is:

$$\begin{array}{cc}
x & P(x) \
4 & 0.08 \
5 & 0.11 \
6 & 0.14 \
7 & 0.19 \
8 & 0.23 \
9 & 0.16 \
10 & 0.09 \
\end{array}$$

What is the probability that at most 6 machines are sold in a day?

Accepted Answer

$$P(x \leq 6) = 0.08 + 0.11 + 0.14 = 0.33$$

Question 10

Which of the following statements about probability is FALSE?

Accepted Answer

The probability of an event can be greater than 1.

Question 11

A game has the following payoffs and probabilities:

$$\begin{array}{cc}
Payoff & Probability \
-10 & 0.2 \
0 & 0.5 \
20 & 0.3 \
\end{array}

Is $$this game fair?

Accepted Answer

No, because $$E(X) = (-10 	imes 0.2) + (0 	imes 0.5) + (20 	imes 0.3) = -2 + 0 + 6 = 4.$$

Question 12

If the probability of finding a defective item is $$p = 0.043$$, how many defectives would you expect in a sample of 300 items?

Accepted Answer

$$0.043 	imes 300 = 12.9$$

Question 13

Which approach to probability is used when you estimate the chance of rain tomorrow based on weather patterns and expert knowledge?

Accepted Answer

Subjective Probability

Question 14

If a die is rolled, what is the classical probability of getting a 4?

Accepted Answer

$$P(4) = \frac{1}{6}$$

Question 15

Which of the following best describes the expected value in a probability distribution?

Accepted Answer

The long-run average value of the random variable if the experiment is repeated infinitely many times.

Question 16

A probability distribution has the following values:

$$\begin{array}{cc}
x & P(x) \
1 & 0.2 \
2 & 0.3 \
3 & 0.5 \
\end{array}$$

What is the mean $$\mu of $$this distribution?

Accepted Answer

$$\mu = 1 	imes 0.2 + 2 	imes 0.3 + 3 	imes 0.5 = 2.3$$

Question 17

Which of the following is an example of empirical probability?

Accepted Answer

Estimating the probability of getting heads by flipping a coin 100 times and recording the results.

Question 18

If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and A and B are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) = 0.4 + 0.5 = 0.9$$

Probability and Discrete Probability Distributions: Study Notes for Introductory Statistics

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