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

You decided to buy 5 math books and are lining them up on yourbookshelf at home. You think it’d be really cool if you had somehowlined them up in order of the date that they were copyrighted. What is the probability of this randomly happening?

What is the probability that one of your two favorites are first in line?

What is the probability that your favorite is first, and your second favorite is second?

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

4 1 13 5

4! 13!* * 5148

3!*1! 8!*5!C C

Yesterday:How many possible 5 card hands are there containing all 5 cards from the same suit?

Today: What is the probability of getting a 5 card hand with all 5 cards being the same suit?

A farmer has 15 empty cages. She has a chicken, a rooster, and a duck. How many different ways can she place these animals into the cages if she can only put one animal in each cage?

What is the probability that someone could guess where she placed each of these animals?

A farmer has 11 empty cages. She has a dog, a Tazmanian devil, a bear, and a armadillo. How many different ways can she place these animals into the cages if she can only put one animal in each cage?

What is the probability that someone could guess where she placed each of these animals?

Be sure to know the difference of finding “the number of different ways” verses “finding the probability”.

Eight balls numbered from 1 to 8 are placed in an bowl. Two balls are selected at random. Find the probability that it is the numbers 3 and 5.

Eight balls numbered from 1 to 15 are placed in an bowl. Three balls are selected at random. Find the probability that it is the numbers 2, 9, 14.

A doctor is researching the effects of a new drug. The test group consists of 40 subjects. 15 subjects are given Drug A, 17 subjects are given Drug B, and 8 subjects are given a placebo. If three subjects are chosen at random, what is the probability that a subject that was given Drug A, a subject that was given Drug B, and a subject that was given a placebo were chosen in that order?

A doctor is researching the effects of a new drug. The test group consists of 50 subjects. 25 subjects are given Drug A, 20 subjects are given Drug B, and 5 subjects are given a placebo. If three subjects are chosen at random, what is the probability that a subject that was given Drug A, a subject that was given Drug B, and a subject that was given a placebo were chosen in that order?

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

Two dice are rolled. What is the probability that the sum is less than four or greater than 9?

A farmer has 15 empty cages. She has a chicken, a rooster, and a duck. How many different ways can she place these animals into the cages if she can only put one animal in each cage?

What is the probability that someone could guess where she placed these animals?

Eight balls numbered from 1 to 8 are placed in an bowl. Two balls are selected at random. Find the probability that it is the numbers 3 and 5.

Eight balls numbered from 1 to 8 are placed in an vase. Two balls are selected at random. Find the probability that one of them is NOT the number 5.

A doctor is researching the effects of a new drug. The test group consists of 40 subjects. 15 subjects are given Drug A, 17 subjects are given Drug B, and 8 subjects are given a placebo. If two subjects are chosen at random, what is the probability that they have both been given Drug A?

A doctor is researching the effects of a new drug. The test group consists of 40 subjects. 15 subjects are given Drug A, 17 subjects are given Drug B, and 8 subjects are given a placebo. If three subjects are chosen at random, what is the probability that a subject that was given Drug A, a subject that was given Drug B, and a subject that was given a placebo were chosen in that order?