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12.3 An Introduction to Probability
What you should learn:
GoalGoal 11
GoalGoal 22
Finding Theoretical and Experimental Probabilities of events.
Finding Geometric probabilities .
12.3 An Introduction of Probability12.3 An Introduction of Probability
THEORETICAL AND EXPERIMENTAL PROBABILITY
The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur.
There are two types of probability: theoretical andexperimental.
THEORETICAL AND EXPERIMENTAL PROBABILITY
THE THEORETICAL PROBABILITY OF AN EVENT
When all outcomes are equally likely, thetheoretical probability that an event A will occur is:
P (A) = total number of outcomes
The theoretical probability of an event is often simply called the probability of the event.
all possible outcomes
number of outcomes in A
outcomes in event A
outcomes in event A
You can express a probability as a fraction, a decimal, or a percent.For example: , 0.5, or 50%.1
2
P (A) = 49
Finding Probabilities of Events
You roll a six-sided die whose sides are numbered from
1 through 6.
Find the probability of rolling a 4.
SOLUTION
Only one outcome corresponds to rolling a 4.
P (rolling a 4) = number of ways to roll a 4
number of ways to roll the die16
=
Finding Probabilities of Events
Three outcomes correspond to rolling an odd number: rolling a 1, 3, or a 5.
P (rolling odd number) = number of ways to roll an odd number
number of ways to roll the die
You roll a six-sided die whose sides are numbered from
1 through 6.
Find the probability of rolling an odd number.
SOLUTION
36
12
= =
Finding Probabilities of Events
All six outcomes correspond to rolling a number less than 7.
P (rolling less than 7 ) = number of ways to roll less than 7
You roll a six-sided die whose sides are numbered from
1 through 6.
Find the probability of rolling a number less than 7.
SOLUTION
number of ways to roll the die66
= = 1
Probabilities Involving Permutations or Combinations
You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song.
What is the probability that the songs are playedin the same order they are listed on the CD?
SOLUTION
There are 8! different permutations of the 8 songs. Of these, only 1 is the order in which the songs are listed on the CD. So, the probability is:
Help
18!
140, 320
P(playing 8 in order) = = 0.0000248
Probabilities Involving Permutations or Combinations
You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song.
You have 4 favorite songs on the CD. What is the probability that 2 of your favorite songs are played first, in any order?
SOLUTION
There are 8C2 different combinations of 2 songs. Of these,
4C2 contain 2 of your favorite songs. So, the probability is:
Help
P(playing 2 favorites first) = = = 0.2144 C 2
8 C 2
628
314
Probabilities Involving Permutations or Combinations
Sometimes it is not possible or convenient to find thetheoretical probability of an event. In such cases youmay be able to calculate an experimental probabilityby performing an experiment, conducting a survey, orlooking at the history of the event.
Finding Experimental Probabilities
In 1998 a survey asked Internet users for their ages. The results are shown in the bar graph.
Finding Experimental Probabilities
SOLUTION
The number of people surveyed was 1636 + 6617 + 3693 + 491 + 6 = 12,443.
Of the people surveyed, 16 36 are at most 20 years old.
So, the probability is:
1636
6617
3693
491
6
P(user is at most 20) = 0.131163612,443
Find the experimentalprobability that a randomly selected Internet user is atmost 20 years old.
Finding Experimental Probabilities
SOLUTION
Find the experimentalprobability that a randomly selected Internet user is atleast 41 years old. Given that 12,443 people were surveyed.
Of the people surveyed, 3693 + 491 + 6 = 4190 are at least 41 years old.
So, the probability is:
P(user is at least 41) = 0.337419012,443
GEOMETRIC PROBABILITY
Some probabilities are found by calculating a ratio oftwo lengths, areas, or volumes. Such probabilities arecalled geometric probabilities.
Using Area to Find Probability
You throw a dart at the board shown. Your dart is equallylikely to hit any point inside thesquare board. Are you more likely to get 10 points or 0 points?
• 32
182= = = 0.0873
3249
36
Using Area to Find Probability
SOLUTION
P (10 points) = area of smallest circlearea of entire board
Are you more likely to get 10 points or 0 points?
You are more likely to get 0 points.
P (0 points) = area outside largest circlearea of entire board
182 – ( • 9 2 )182
= = = 0.215324
324 – 814
4 –
Assignment
