Sec. 1.6 Probability

Objective: Students set up probability equations appropriately

Experimental Probability

Probability of event =

Number of times event occurs

Number of trials

Example 1

A player hit the bull’s eye on a circular dartboard 8 times out of 50. Find the experimental probability that the player hits the bull’s eye.

We need to use the formula.

Number of times event occurs =

Number of trials

816%

50

Example 2

Find the theoretical probability of rolling a multiple of 3 with a number cube? How about rolling an odd?

The Cube is a normal six sided di.

A) How many numbers on the cube are a multiple of 3?

Yes 2 numbers, 3 and 6. So we get. 2 = 1 6 3B) How many odds?Yes 3 numbers, 1,3,5So we get 3 = 1 6 2

Example 3 Suppose that all the points on the circular

dartboard shown below are equally likely to be hit by a dart you have thrown. Find the probability of only scoring 2 points with one throw.

Note: The radius of each circle is one unit larger than the one below it. 2020

1052

First we need to find the area of the whole dart board. This is the denominator because any throw can hit any where on the dart board.

To find the area of the green we need to subtract the areas of the others. So we get (using area πr2 of a circle)

π(4r)2 – π(3r)2

π(4r)2

= 16πr2 - 9πr2

16πr2 = 7πr2 16πr2

202010

52

7

16

Evaluate the expression

1) 3! 2) 7!

3) ( 4!/6! )

= 3 • 2 • 1

= 6

= (4 • 3 • 2 • 1) / (6 • 5 • 4 • 3 • 2 • 1)

= 7 • 6 • 5 • 4 • 3 • 2 • 1

= 5040

= 1/(6 • 5) = 1/30

Objective – To be able to use permutations to count the number of ways an event can happen

State Standard –

18.0 Students use the fundamental counting principles to compute combinations and permutations.

19.0 Students use Combinations and Permutations to compute probabilities.

The Fundamental Counting Principle

Suppose you own a small deli. You offer 4 types of meat (ham, turkey, roast beef, and pastrami) and 3 types of bread (white, wheat, and rye). How many choices do your customers have for a meat sandwich?

Ham

Turkey

Roast Beef

Pastrami

White

Wheat

RyeWhite

Wheat

RyeWhite

Wheat

Rye White

Wheat

Rye

Because you have 4

meats and 3 breads:

The Total # of choices is

4 • 3 = 12

Pastrami on White

Pastrami on Wheat

Pastrami on Rye

Example 1a

At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12 different drinks, and 6 different desserts. How many different dinners consisting of 1 salad, 1 entrée, 1 drink, and 1 dessert can you choose?

8 • 2 • 12 • 6 =

1152 different dinners

Example 1b

How many different 7 digit phone numbers are possible if the first digit cannot be 0 or 1?

8

1st d

igit

2nd d

igit

• 10

3rd d

igit

• 10

4th d

igit

5th d

igit

6th d

igit

7th d

igit

• 10 • 10 • 10 • 10 = 8,000,000

A Permutation – is an ordering of n objects.

For instance, there are 6 permutations of the letters A, B, and C:

ABC, ACB, BAC, BCA, CAB, CBAIn general there are 3 choices for the 1st letter, 2 choices for the 2nd letter, and 1 choice for the 3rd letter so there are:

3 • 2 • 1 =6 ways to arrange the letters.n!

Permutations of n objects taken r at a time

The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr and is given by:

nPr = n!/(n – r)!

Example 2

Find the number of permutations.

5 P2

nPr = n!/(n – r)!

5 P2 = 5!/(5 – 2)!

= 5 • 4 • 3 • 2 • 1 3 • 2 • 1

= 5 • 4 = 20

Find the Product:

1) (2x – 3)2 2) (x – 3y)2 = (2x – 3)(2x – 3)

= 4x2 – 12x + 9

= (x – 3y)(x – 3y)

= x2 – 6xy + 9y2

Objective – To be able to use combinations to count the number of ways an event can happen

State Standard – 19.0 Students use Combinations and Permutations to compute probabilities.

Combinations of n objects taken r at a time

The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr and is given by:

nCr = n!

(n – r)! • r!

Example 3a

Find the number of combinations.

5 C2

nCr = n!/{(n – r)! • r!}

5 C2 = 5!/{(5 – 2)! • 2!}

= 5 • 4 • 3!

3! • 2!

= 5 • 4 2

= 10

Example 3b

Find the number of combinations.

10 C4

nCr = n!/{(n – r)! • r!}

10 C4 = 10!/{(10 – 4)! • 4!}

= 10 • 9 • 8 • 7 • 6!

6! • 4!

= 10 • 9 • 8 • 7

4 • 3 • 2 • 1= 210

3

Example 4

Use a standard deck of 52 cards.

a) If the order is not important, how many different 7-card hands are possible?

=133,784,560

b) How many of these hands have exactly 6 cards of the same suit?

6,864

52C7 = 52!

(52 – 7)! • 7!

4C1 • 13C6 =

When finding the # of ways both an event A and an event B can occur you need to multiply.

A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles, and 12 red marbles. Suppose you pick one marble at random. Find each probability.

1) P(yellow) 2) P(not blue)

3) P(green or red)

14

72

50

72

7

36

1

2

25

36

36

72

Objective – To be able to find the probabilities of events A and B, and of events A or B.

State Standard –

1.0 Students will know the definition of the notion of independent events and can use the rules of addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.

Dependent Events – Is when the outcome of one event affects the outcome of a second event.

Independent Events – Is when the outcome of one event does not affect the outcome of a second event.

Example 1

Classify each pair of events as dependent or independent.

a) Toss a coin. Then, select a marble from a bag that contains marbles of different colors.

b) Select a marble from a bag that contains marbles of two different colors. Put the marble aside, and select a second marble from the bag.

Independent

Dependent

Checking for Understanding:

Suppose you select a marble from a bag of marbles. You replace the marble and then select again. Are your selections dependent or independent event? Explain.

Independent; The number of marbles is the same after the marble is replaced.

If A and B are independent events, the probability that both A and B occur is:

P(A and B) = P(A)·P(B)

Probability of A and B

Example 2:

A random number generator on a computer selects three integers from 1 to 20. What is the probability that all three numbers are less than or equal to 5?

The probability of selecting a number from 1 to 5 is:

5( )

20P A

1 1 1( ) ( ) ( )

4 4 4P A P A P A

1

64

So the probability that all three numbers will be less than or equal to 5 is:

1

4

Example 3:

A person draws 2 cards from a deck of 52 cards. What is the probability that the 2 cards drawn will both be face cards if the deck contains 12 face cards?

12( )

52P A 11

( )51

P B

12 11( ) ( )

52 51P A P B

132 11

2652 221 = 0.05

MATHPOWERTM 12, WESTERN EDITION

12.2

8.4.1

Probability

8.4.2

Conditional Probability

If A and B are events from an experiment, the conditional probability of B given A (P(A|B)), is the probability that Event B will occur given that Event A has already occurred. The conditional probability is equal to the probability that B and A will occur divided by the probability that B will occur.

This is given in Bayes’ Formula:

P(A | B)

P(B and A)P(B)

, where P(B) 0

8.4.3

Conditional Probability

Determine the conditional probability for each of the following:

a) Given P(B and A) = 0.725 and P(B) = 0.78, find P(A|B).

P(A | B)

P(B and A)P(B)

P(A | B)

0.725

0.78

P(A|B) = 0.9295

a) Given P(blonde and tall) = 0.5 and P(blonde) = 0.73, find P(A|B).

P(A | B)

P(B and A)P(B)

P(A | B)

0.5

0.73P(A|B) = 0.6849

The table shows the results of a class survey. Find P(own a pet | female)

The condition female limits the sample space to 14 possible outcomes.

Of the 14 females, 8 own a pet.

Therefore, P(own a pet | female) equals . 8 14

yes nofemale 8 6male 5 7

Do you own a pet?

Conditional ProbabilityLESSON 12-2

Additional Examples

Using the data in the table, find the probability that a sample of

non-recycled waste was plastic.

The given condition limits the sample space to non-recycled waste.

Material Recycled Non-RecycledPaper 36.7 45.1Metal 6.3 11.9Glass 2.4 10.1Plastic 1.4 24.0Other 21.2 70.1

The probability that the non-recycled waste was plastic is about 15%.0.15

A favorable outcome is non-recycled plastic.

P(plastic | non-recycled) = 24.0 45.1 + 11.9 + 10.1 + 24.0 + 70.1

Conditional ProbabilityLESSON 12-2

Additional Examples

= 24.0 161.2

Researchers asked people who exercise regularly whether they jog or walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a male respondent jogs.

Relate: P( male ) = 58%P( male and jogs ) = 20%

Define: Let A = male.Let B = jogs.

The probability that a male respondent jogs is about 34%.

Write: P( A | B ) =P( A and B )

P( A )

= Substitute 0.2 for P(A and B) and 0.58 for P(A).

0.344 Simplify.

0.2 0.58

Conditional ProbabilityLESSON 12-2

Additional Examples

A student made the following

observations of the weather in his

hometown.

Conditional ProbabilityLESSON 12-2

Additional Examples

Use a tree diagram to find the probability that a day will start out clear, and then it will rain.

During the mostly clear days, it rained 4% of the time.

•

On 28% of the days, the sky was mostly clear.

•

During the cloudy days it rained 31% of the time.

•

P(norain|cloudy)

P(norain|clear)

P(rain|cloudy)

P(rain|clear)

(continued)

Conditional ProbabilityLESSON 12-2

Additional Examples

The path containing clear and rain represent days that start out clear and then will rain.

P(clear and rain) = P(rain | clear) • P(clear)= 0.04 • 0.28= 0.011

The probability that a day will start out clear and then rain is about 1%.

12.4 Warm Up

Find the mean, median and mode of the data set.

1) 6, 7, 9, 9, 9,10

Mean: x = 6+7+9+9+9+10 = 50 = 8.33 6 6

Median: 9

Mode: 9

12.4 Standard Deviation

Standard Deviation

standard deviation- is often very important in scientific experiments. A small standard deviation signifies the results are consistent. A large standard deviation signifies the results are inconclusive.

The standard deviation (read as “sigma”) of x1,x2…. xn

= (x1 – x )2+(x2 – x )2+…+(xn - x)2 n

Example 1

What is the S.D. of the 3 test scores?

86, 89, and 91

= (x1 – x )2+(x2 – x )2+…+(xn - x)2 n

Mean: x = 86 + 89 + 91 = 88.67 3

= 2.06

Example 2

What is the S.D. of the 4 test scores?

70, 85, 99 and 40

= (x1 – x )2+(x2 – x )2+…+(xn - x)2 n

Mean: x = 70 + 85 + 99 + 40 = 294 = 73.5 4 4

= 21.89