$$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$

$$M = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Question 1

How many ways can you reorder 5 unique books on a shelf?

Accepted Answer

$$5! = 5 	imes 4 	imes 3 	imes 2 	imes 1 = 120$$

Question 2

Which formula would you use to determine the number of ways to select 3 students from a group of 10, where order does NOT matter?

Accepted Answer

$$nCr = \frac{n!}{(n-r)!r!}$$

Question 3

A password consists of 4 letters, and letters can repeat. How many unique passwords are possible using the 26-letter English alphabet?

Accepted Answer

$$26^4$$

Question 4

Which of the following best describes a discrete probability distribution?

Accepted Answer

It assigns probabilities to a finite or countable set of outcomes.

Question 5

Given a random variable $$X$$ with possible values 1, 2, and 3, and $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3$$, what is the expected value $$M$$?

Accepted Answer

$$M = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 6

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

Accepted Answer

The sum of all probabilities must be less than 1.

Question 7

If you have 7 identical red balls and 3 identical blue balls, how many unique ways can you arrange all 10 balls in a row?

Accepted Answer

$$\frac{10!}{7!3!}$$

Question 8

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.7$$, $$P(X=1)=0.2$$, $$P(X=2)=0.1. $$What is $$P(X \geq 1)$$?

Accepted Answer

$$P(X \geq 1) = 0.2 + 0.1 = 0.3$$

Question 9

Which formula correctly calculates the variance $$\$$sigma^2$$ of a discrete random variable X$?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - M)^2$ P(x)$$

Question 10

A lottery has 100 tickets, and only 1 ticket wins. If you buy 5 tickets, what is the probability you win?

Accepted Answer

$$\frac{5}{100} = 0.05$$

Question 11

Which of the following is an example of a continuous probability distribution?

Accepted Answer

The probability of a randomly chosen person having a height between 160 cm and 170 cm.

Question 12

If $$P(X \leq x) \u003c 0.05$$, what does this typically indicate about the value $$x in a $$probability distribution?

Accepted Answer

The value $$x is $$unusually low (significantly lower than expected).

Question 13

Which of the following best defines a random variable?

Accepted Answer

A variable whose value is determined by chance.

Question 14

If $$P(X \geq x) \u003c 0.05$$, what does this typically indicate about the value $$x in a $$probability distribution?

Accepted Answer

The value $$x is $$unusually high (significantly higher than expected).

Question 15

Which of the following is the correct formula for the expected value $$E of a $$discrete random variable $$X$$?

Accepted Answer

$$E = \sum x P(x)$$

Permutations, Combinations, and Discrete Probability Distributions

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