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Unit 4: Probability
Probability is a branch of mathematics in which a numerical value is used to express the likelihood of an event,
or outcome.
 An outcome is the result of some activity. Probability has to do with certainty, uncertainty, and
prediction. In other words, is has to do with chance.
 The probability of an event is the chance that it will occur, expressed as a ratio of a specific event to all
possible events. All the possible results, or outcomes, are called the sample space.
Sample Space
A sample space is a set of all possible outcomes for an activity or experiment.
Rolling a die
Sample Space
{1, 2, 3, 4, 5, 6}
Tossing a coin
{ Heads, Tails}
Drawing a card from a standard deck
{52 cards}
Drawing one marble from the bottle
{8 marbles}
Rolling a pair of dice
Choosing an outfit from a green blouse, a red
blouse, a black skirt, a pair of sneakers, and a
pair of sandals
{(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)}
{green blouse-skirt-sneakers,
green blouse-skirt-sandals,
red blouse-skirt-sneakers,
red blouse-skirt-sandals}
A tree diagram is a visual representation of the possible outcomes of a probability problem.
Example 1: Suppose you wish to list all possible outcomes of a coin toss followed by a roll of a die using a tree
Example 2: You are at a carnival. One of the carnival games asks you to pick a door and then pick a curtain
behind the door. There are 3 doors and 4 curtains behind each door. How many choices are possible for the
Example 3: A restaurant sells kids' meals consisting of one main course, one side dish, and one drink, as
shown in the table below.
a) Draw a tree diagram or list the sample space showing
all possible kids' meals.
b) How many different kids' meals can a person order?
c) Jose does not drink juice. Determine the number of different kids' meals that do not include juice.
d) Jose's sister will eat only chicken nuggets for her main course. Determine the number of different kids'
meals that include chicken nuggets.
There must be a short cut to help you count the number of outcomes! Can we come up with it? It’s called…
the Counting Principle!
Roll a die and flip a coin
# of Possibilities
Draw two cards from a standard deck of 52 cards
without replacement
If ice cream sundaes come in 5 flavors with 4
possible toppings, how many different sundaes
can be made with one flavor of ice cream and one
A coin is tossed five times
A die is rolled four times
Another example:
How many different sandwiches consisting of one type of cheese, one condiment, and one bread choice can be
prepared from five types of cheese, two condiments, and three bread choices?
One more example: Your state issues license plates consisting of letters and numbers. There are 26 letters and
the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license
plates can be issued with two letters followed by three numbers?
Probabilities can be written as a fraction, as a percent, or as a decimal. The probability of an event happening
is always greater than or equal to zero and less than or equal to one. [0, 1]
If an event cannot happen, the probability is 0.
If an event is certain to happen, the probability is 1.
Likelihood of Outcome
Rolling a 2
P(2) =
Rolling a 7
P(2) =
Rolling a number less than 10
P(number less than 10) =
Rolling a factor or 6 (1,2,3,6)
P(factor of 6) =
Rolling a number less than 3
P(number less than 3) =
Rolling a number that is an integer
P(integer) =
Simple Event: rolling a fair die
1) List the possible outcomes (sample space):
2) P(3) =
3) P(a number less than 3) =
4) P(an odd number) =
5) P( prime number) =
6) P(factor of 12) =