Probability
In any random probability problem, it is essential to identify all the different outcomes that could occur.
Definitions of terms that may be used in a probability problem
Experiment - a situation that involves chances that lead to results called outcomes.
Outcome - the result of an individual try or trial in an experiment.
Event - one or more of the outcomes in an experiment.
Probability - the measure of how likely an event can be.
Formula to finding probabilities
To accurately identify and measure the probability of an event, the following formula is used:
The Probability of an Event =
Then Number of Ways an Event can Occur
The Total Number of Possible Outcomes
Example 1
A spinner has 4 equal sections colored yellow, blue, green and red. If the spinner is turned, what is the
probability of each event:
¾ landing on yellow =
¾ landing on blue =
number of ways to land on yellow
= 14
total number of different colors
number of ways to land on blue
= 14
total number of different colors
¾ landing on blue or red =
number of ways to land on blue or red
= 2 4 = 12
total number of different colors
¾ landing on blue , red or green =
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number of ways to land on blue , red or green
= 34
total number of different colors
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Example 2
A standard die is rolled. What is the probability of each event?
¾ rolling a one =
number of ways to roll a one
total number of sides
¾ rolling a six =
number of ways to roll a six
total number of sides
¾ rolling an even number =
¾ rolling an odd number =
= 16
= 16
number of ways to roll an even number
total number of sides
number of ways to roll an odd number
total number of sides
= 3 6 = 12
= 3 6 = 12
Example 3
A container has 7 blue, 6 red, 10 yellow and 9 green balls inside of it. If you pick a single ball at
random without looking, what is the probability that you will choose each given event?
¾ picking a red ball =
¾ picking a blue ball =
number of ways to pick a red ball
= 6 32 = 316
total number of balls
number of ways to pick a blue ball
= 7 32
total number of balls
¾ picking a red or blue ball =
number of ways to pick a red or blue ball
= 13 32
total number of balls
¾ picking a blue, green or red ball =
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number of ways to pick a blue, green or red ball 22
= 32 = 1116
total number of balls
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The sum of dice problem
When given a question on the sum of dice, a table is required to find the probability. Like the
multiplication table, the sum table can be created to see the various types of sums between the dice.
(die 1, die2)
1
2
3
4
5
6
1
2
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
3
4
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
5
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
6
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
Taking the sum of each scenario.
(die 1, die2)
1 2 3
4
5
6
1
2
2 3 4
3 4 5
5
6
6
7
7
8
3
4 5 6
7
8
9
4
5 6 7
8
9
10
5
6
6 7 8 9 10 11
7 8 9 10 11 12
Example 1
Given two dice, what is the probability of having a sum greater than or equal to 7?
Overall there are 36 possible outcomes; the chart has six 7’s, five 8’s, four 9’s, three 10’s, two 11’s, and
one 12. There are 21 sums that are greater than or equal to 7. Using the formula:
21 7
Pr (sum ≥ 7 ) =
=
36 12
Example 2
Given two dice, what is the probability of having a sum that is greater than or equal to 3 and less than 6?
The probability of having a sum greater than or equal to 3 and less than 6 would be written as sum ≥ 3 and
sum < 6 , the inequality can be written 3 ≤ sum < 6 . There are two 3’s, three 4’s, four 5’s, so
9 1
Pr (3 ≤ sum < 6) =
= .
36 4
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