Basic Probabilities
Starting Unit 6 Today!
Definitions
 Experiment – any process that
generates one or more observable
outcomes
 Sample Space – set of all possible
outcomes of the experiment
Examples
Experiment
Sample Space
Tossing a coin
Head and tails, written as
{H, T}
Rolling a dice
{1, 2, 3, 4, 5, 6}
Choosing a name from a
phone book
All the names in the phone
book
Counting the number of fish
in Lake Norman
The set of non-negative
integers
Definitions
 Event – any outcome or set of outcomes in the
sample space

For example, in the experiment of rolling a dice, the
set {1, 2, 5} is an event…meaning you rolled a 1, a
2, and a 5.
 Probability – how likely the event is to occur


A probability of 0 means the even cannot occur
A probability of 100 means the even must occur
Probability Distribution
 A table describing all probabilities from
the experiment.
 The sum of all probabilities should equal
1 or 100% of the data.
Probability Distributions
 Suppose that 100 marbles are placed in a bag; 50 red,
30 blue, 10 yellow, and 10 green. An experiment
consists of drawing one marble out of the bag and
observing the color.



What is the sample space?
Write out a reasonable probability distribution for this
experiment and verify that the sum of the probabilities is 1.
What is the probability that a blue or a green marble will
be drawn?
Mutually Exclusive
 Two events are mutually exclusive if they
have no outcomes in common.


For example, the probability of rolling an even
number and the probability of rolling a 1.
P(E or F) = P(E) + P(F)
 The compliment of an event is the probability
that it doesn’t happen

1-probability that it happens
Example
a)
Which of the following pairs of events E
and F are mutually exclusive?



b)
c)
Out- Prob.
come
A
0.4
E = {A, C, E}
E = {a vowel}
E = {a vowel}
F = {C, S}
F = {in the first 4
letters of the
alphabet}
F = {C}
What is the compliment of the event
{A, S}?
What is the probability of the event “the
spinner does not land on A?”
S
0.3
C
0.2
E
0.1
Independent Events
 Two events are independent if the
occurrence or non-occurrence of one
event has no effect on the probability of
another event.

P(E and F) = P(E) · P(F)
Mutually Exclusive v.
Independent
Mutually Exclusive
Refers to two possible
results for a single trial of
an experiment.
Independent
Refers to the results from
two or more trials of an
experiment.
OR
AND
P(E or F) = P(E) + P(F)
P(E and F) = P(E) · P(F)
Example
 The probability of winning a certain
game is 0.1. Suppose the game is
played on two different occasions. What
is the probability of:

Winning both times?

Losing both times?

Winning once and losing once?
Random Variables
 A random variable is a function that
assigns a number to each outcome in
the same space of an experiment.

For example, heads = 1, tails = 0
Example
 An experiment consists of rolling 2 dice.
A random variable assigns to each
outcome the total of the faces shown.



Write out the same space for the
experiment.
Find the range of the random variable
List the outcomes to which the value 7 is
assigned.
Answer

 The smallest value is 2, which is assigned to
(1,1) and the highest is 12 (6, 6). The range is
the set of integers from 2 to 12.
 The value of 7 is assigned to (1, 6), (2, 5),
(3, 4), (4, 3), (5, 2), and (6, 1)
Expected Value
 The expected value, or mean, of a
random variable is the average value of
the outcomes.
 As the experiment is tested a large
number of times the average will be the
expected outcome.

For example, flipping a coin…the more
times you do it the closer your results are
to 50% heads and 50% tails.
Example
 A probability distribution for
a $1 instant-win lottery ticket
is given below. Find the
expected value and interpret
the results.
Win
$0
$3
$5
$10
$20
$40
$100
$400
$2500
Prob.
.882746
.06
.04
.01
.005
.002
.0002
.00005
.000004
Homework
 Worksheet (1-24)