Question

Practice

1. Suppose you roll a number cube once.
a. Identify the sample space.
b. What is the probability of rolling a 5, P(5)?
c. What is the probability of rolling an even number, P(even)?
d. What is the probability of rolling a number greater than 2, P(greater than 2)?
e. Construct the probability model for rolling a number cube.
Outcomes
Probability
f. Is the probability model from part (e) a uniform or non-uniform probability model? Explain
your reasoning.
g. What is the probability of rolling a number that is not a multiple of 3, P(not a multiple of 3)?

Answer 1

Part A

Answer: sample space is {1,2,3,4,5,6}

Reason: It is the set of possible outcomes, i.e. possible die rolls.

=======================================================

Part B

Answer: 1/6

Reason: There is one face showing "5" out of 6 faces total.

=======================================================

Part C

Answer: 1/2

Reason:

Half of the numbers are even.

Three numbers are even out of 6 total. So 3/6 = 1/2.

=======================================================

Part D

Answer: 2/3

Reason:

The outcomes we want are {3,4,5,6} which are 4 outcomes out of 6 total. Therefore we get 4/6 = 2/3 as the probability of getting something larger than 2.

=======================================================

Part E

This is what the probability model looks like

`\begin{array}c \cline{1-7}\text{Outcomes} & 1 & 2 & 3 & 4 & 5 & 6\\\cline{1-7}\text{Probability} & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6\\\cline{1-7}\end{array}`

The first row deals with the sample space. Refer to part A to see the sample space. It's the list of all possible outcomes on the number cube.

In the bottom row is the probability for each outcome. Assuming a fair number cube, each probability should be 1/6, since we have 1 side out 6 total. Each side has equal chance of showing up on any given roll.

Notes:

• Everything in the bottom row adds to 1
• The term "probability model" is interchangeable with "probability distribution".
• This distribution is discrete because of the finitely many outcomes

=======================================================

Part F

Answer: Uniform

Reason:

A uniform probability distribution has each outcome with the same chances of it happening. That probability being 1/6 in this case.

Another example of a uniform distribution would be a coin toss. Each outcome (heads or tails) has probability 1/2.

Think "uniform" as in "same", in that every solider in the army wears the same uniform.

=======================================================

Part G

Answer: 2/3

Reason:

The multiples of 3 on the number cube are {3,6}

The non-multiples of 3 are {1,2,4,5}. We have 4 outcomes we want out of 6 total, so we get to the answer of 4/6 = 2/3. Interestingly this is the same fraction as part D.