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An introduction to probability theory, focusing on the fundamental counting principle, permutations, and combinations. It covers the concepts of trials, experiments, sample spaces, events, and probability, as well as the difference between theoretical and experimental probabilities. The document also includes examples of calculating probabilities using permutations and combinations, and explains the difference between these two concepts. Students studying statistics, mathematics, or probability theory will find this document useful for understanding the fundamental concepts of probability and counting.
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In this unit you will begin by learning the fundamental counting principle and applying it to probabilities. You will then explore permutations, which are used when the outcomes of the event(s) depend on order, and combinations, which are used when order is not important.
Introduction to Probability
Permutations
Calculate the Number of Possible Outcomes
Combinations
Probability is the likelihood of an event occurring.
Terminology (a coin is used for each of the examples) Definition Example Trial : a systematic opportunity for an event to occur tossing a coin in the air
Experiment : one or more trials tossing a coin 6 times
Sample space : the set of all possible outcomes of an event H or T
Event : an individual outcome or any specified combination of outcomes. landing H or landing T
Probability is expressed as a number from 0 to 1. It is written as a fraction, decimal, or percent.
The probability of an event can be assigned in two ways:
1.) experimentally : approximated by performing trials and recording the ratio of the number of occurrences of the event to the number of trials. (as the number of trials in an experiment increases, the approximation of the experimental probability increases).
2.) theoretically: based on the assumption that all outcomes in the sample space occur randomly.
Permutations
A permutation is an arrangement of objects in a specific order. When objects are arranged in a row, the permutation is called a linear permutation.
Example #1 : On a baseball team, nine players are designated as the starting line up. Before a game, the coach announces the order in which the nine players will bat. How many different orders are possible?
9! = 9 8 7 6 5 4 3 2 1⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= 362, 880 possible orders.
*Note: When the coach is choosing, on his first choice he has nine players to choose from. Once he makes that choice, he the has eight players left to choose from, then seven, then six, and so on.
If you want to use your calculator to find 9! Press 9, MATH , move the cursor over to PRB, and go down to 4:! Then press ENTER.
Permutations of n Objects
The number of permutations of n objects is given by n! (! is called factorial and means to multiply all consecutive natural numbers starting with n ).
4! = 4 3 2 1⋅ ⋅ ⋅ = 24
Permutations of n Objects Taken r at a Time
The number of permutations of n objects taken r at a time, denoted by
Example #2 : Find the number of ways to listen to 6 different CD’s from a selection of 18 CD’s.
18 6
Note: Since the order in which the CD’s will be played is important, this is a “permutation” problem.
Now we are ready for permutations.
Example #4 : How many ways can the letters of the word “random” be arranged?
This example requires a permutation. It’s formula is P n n ( , ) = n !where we are selecting all of the letters in the arrangement.
( , )! (6, 6) 6! (6, 6) 6 5 4 3 2 1 (6, 6) 720
P n n n P P P
There are 720 ways the letters in the word “random” may be arranged.
Example 5 : If we look at arranging letters in the word “success”, we need to realize that when an s or c is selected, it does not matter which is which. So there are less ways to select the arrangement.
This is called a permutation with repetition and is given by the following formula ! !!
n P a b
= where “ a ” and “ b ” are repeating letters.
How many ways are there to arrange the letters in the word success?
We are using all 7 letters but the “s” has 3 repeats and the “c” has 2 repeats.
2
The letters in the word “success” may be arranged 420 different ways.
An arrangement of objects in which order is not important is called a combination.
Example #1 : How many ways are there to give 4 honorable mention awards to a group of 10 students?
3
10 4
-cancel the 6!'s 4!(6!)
10 9 8 7 4 3 2 1
210
Note: Since the order in which the honorable mention awards are presented is not important, then this is a “combination” problem.
Combinations of n Objects Taken r at a Time
The number of combinations of n objects taken r at a time is given by:
, where
, and have the same meaning.
All are read " choose ".
n n r r
n n r r
