probability – the likelihood that an event will occur. probability is usually expressed as a real...

Probability – the likelihood that an event will occur.

Probability is usually expressed as a real number from 0 to 1.

The probability of an impossible event is 0.

The probability of a certain event is 1.

Experimental Probability – Probability that is determined by collecting data from oberservations

P(event) = # of times the event occurs# of trials

Example of Experimental Probability…

A baseball player got a hit 21 times in 60 at-bats. Find the experimental probability of his getting a hit.

2160

=720

Theoretical Probability – Probability based on a sample space that has equally likely outcomes.

P(event) = Favorable outcomesPossible outcomes

Sample Space – All possible outcomes.

Example of Theoretical Probability…

Find the theoretical probability of getting an even number when you roll a number cube.

36

= 12

Examples…

You select a number at random from the sample space {1,2,3,4,5}.

15

45

35

Find P(2).

Find P(<5).

Find P(prime).

Answer

Answer

Answer

Is this an example of theoretical

or experiment

al probability?

Examples…

In a class of 19 students, 10 study Spanish, 7 study French, and 2 study both Spanish and French. One student is picked at random.

919

0

219

Find P(neither).

Find P(both).

Find P(French).

Answer

Answer

Answer


or experiment

al probability?

Examples…In a telephone survey of 150 households, 75 respondents answered “Yes” to a particular question, 50 answered “No”, and 25 were “Not Sure”.

75150

=12

Find P(Yes).

Find P(not Not Sure).

125150 =

56

Answer

Answer


or experiment

al probability?

Examples…A wallet contains four bills with denominations of $1, $5, $10, and $20. You choose two of the four bills from the wallet at random and add the dollar amounts.What is the sample space.

1 1

1 5

1 10

1 20

5 1

5 5

5 10

5 20

10 110 510 1010 2020 120 520 1020 20

2

6

11

21

10

15

25

20

30

40

A wallet contains four bills with denominations of $1, $5, $10, and $20. You choose two of the four bills from the wallet at random and add the dollar amounts.

0

310

Find P($50).

Find P(at least $25).

Answer

Answer

$2

$6

$11

$21

$10

$15

$25

$20

$30

$40

Probability of Multiple Events

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

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

Let’s Classify Events

Roll a number cube. Then toss a coin.

Pick a flower from a garden. Then pick another flower from the same garden.

Dependent

Independent

Probability of A and B– If A and B are independent, then

P(A and B) = P(A)P(B)

Example…A jar contains 6 blue marbles and 8 yellow marbles. Find the probability of selecting a yellow marble and then tossing a coin and getting tails.

Independent

P(yellow and tails) = P(yellow)P(tails)P(yellow and tails) = 8

1412

P(yellow and tails) =27

A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times?

125

A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.

112

Mutually Exclusive Events- when two events cannot happen at the same time. (they have no outcomes in common).

Non-mutually Exclusive Events- when two events can happen at the same time. (they have one or more outcomes in common).

Mutually Exclusive or Not…

A single 6-sided die is rolled. What is the probability of rolling an odd number or an even number?

A single 6-sided die is rolled. What is the probability of rolling a 5 or an odd number?

Mutually exclusive…they cannot occur at the same time.

Not mutually exclusive…

they can occur at the same time.

Your turn…Is the following scenario mutually exclusive?

A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a 5 or a king?

YesAnswer


A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a club or a king?

NoAnswer


A single letter is chosen at random from the word SCHOOL. What is the probability of choosing an S or an O?

YesAnswer

Determining Probability of Mutually Exclusive Events

When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B)


A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green?P(Yellow or Green) = P(Yellow) + P(Green)

P(Yellow or Green) = +

410

310

P(Yellow or Green) =710


A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?

P(2 or 5) = P(2) + P(5)

P(2 or 5) = +16

16

P(2 or 5) = 13

12


A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the probability of landing on red or blue after spinning this spinner?

Answer

Determining Probability of Non-Mutually Exclusive Events

When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) - P(A and B)

In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?


P(girl or A student) = P(girl) + P(A) - P(girl and A)P(girl or A student) = + -

1330

930

530

1730

P(girl or A student) =

.26

On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the probability of a person driving while intoxicated or having a car accident?


Answer

Dependent ProbabilityWhen two events, A and B,

are dependent, the probability of both occurring

is: P(A and B) = P(A) · P(B|A)

In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested?

Dependent ProbabilityMr. Burger needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly chooses one student who comes to the front of the room. He then chooses a second student from those still seated. What is the probability that both students chosen are girls?

P(student 1 and student 2) = P(1) · P(2|1)

P(student 1 and student 2) = · 1729

1830

P(student 1 and student 2) = 306

870

Dependent ProbabilityIn a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested?

6Answer

probability – the likelihood that an event will occur. probability is usually expressed as a real...

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