This document discusses probability concepts including events, outcomes, sample spaces, and determining probabilities of events. It provides examples of determining the number of elements in events using tree diagrams and the "two dice box" method. It also discusses experiments with equally likely outcomes and how to calculate the probabilities of events for such experiments. Practice problems are included throughout with explanations and solutions.

1. 2.1
Probabilities, Counting, and Equally Likely Outcomes

2. Events
 An event is a subset of the sample space of an
experiment.
 Ex: The event of an “even number” from the
experiment of rolling a die.
 Sample space: {1, 2, 3, 4, 5, 6}
 “qualifying elements”: {2, 4, 6}

3. Quiz 2.1 #1
 Consider the experiment of flipping a coin
twice. How many elements are in the event of
“flipping at least one head?” (Hint: draw a tree
diagram [1.4] and determine the sample space
first, then determine which elements “qualify”)
A. 2
B. 3
C. 4

4. Quiz 2.1 #1
 Consider the experiment of flipping a coin
twice. How many elements are in the event of
“flipping at least one head?” (Hint: draw a tree
diagram [1.4] and determine the sample space
first, then determine which elements “qualify”)
A. 2
B. 3
C. 4
 Answer: B

5. Two Dice “Box” Method
 Visual representation of the sample space of two
dice roll:

6. Two Dice Box Example
 Suppose we want to know the number of elements in the event
“sum less than or equal to 4.” Here’s how we use this box
(Circles represent qualifying elements): First die
Second
die
 Therefore, the answer is 6.

7. Quiz 2.1 #2
 Consider the experiment of rolling two dice.
What is the number of elements in the event
“difference between rolls is at least 3?”
A. 12
B. 14
C. 16

8. Quiz 2.1 #2
 Consider the experiment of rolling two dice.
What is the number of elements in the event
“difference between rolls is at least 3?”
A. 12
B. 14
C. 16
 Answer: A

9. Outcomes and Probabilities
 For any experiment, each outcome is said to
have a “probability” or “weight” – the likelihood
of that event compared to other ones.
 The probability of all possible outcomes of an
experiment must sum up to 1.

10. Equally Likely Outcomes
 For some experiments, it is intuitive that all
outcomes of the experiment are equally likely.
 For example, the outcomes {1, 2, 3, 4, 5, 6} from
rolling a “fair” die is equally likely.
 Since the probabilities have to sum up to one,
each element has a probability of 1/6.

11. Weighted Probabilities
Let’s consider the following experiment:
 An urn has 2 red, 1 white, and 1 blue balls.
 Let O1 = red, O2 = white, O3 = blue.
 O means Outcome
 Since the chance of drawing each ball is equally
likely, each ball has ¼ chance of being drawn
w1 = .5, w2 = .25, w3 = .25
 W for weights
 w1 + w2 + w3 = 1

12. Quiz 2.1 #3

13. Quiz 2.1 #3
 Let consider an experiment of drawing a card
from a deck of cards. What’s the probability of
drawing an Ace?
A. 1/12
B. 1/13
C. 1/52

14. Quiz 2.1 #3
 Let consider an experiment of drawing a card
from a deck of cards. What’s the probability of
drawing an Ace?
A. 1/12
B. 1/13
C. 1/52

15. Quiz 2.1 #3
 Let consider an experiment of drawing a card
from a deck of cards. What’s the probability of
drawing an Ace?
A. 1/12
B. 1/13
C. 1/52
 Answer: B

16. Summary
 Definition:
 event
 outcome, weight
 How to determine the number of elements in an
event
 How to use “Two Dice Box”
 Equally likely outcomes
 Determining probabilities of events with an
experiment containing equally likely outcomes.

17.  Features
 27 Recorded Lectures
 Over 116 practice problems with recorded solutions
 Discussion boards/homework help
 Visit finitehelp.com to find out more
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