Math I

Probability is the chance that something will happen.

Probability is most often expressed as a fraction, a decimal, a percent, or can also be written out in words.

To determine the probability…

P(event) =           number of true outcomes                           total number of equally likely outcomes

Independent: Two events are independent if the occurrence of one has no effect on the occurrence of the other…

Example 1: (probability of two events)What is the probability of drawing a king and then an ace from a standard 52 card deck with replacement?

P(King, Ace) =

Example 2:What is the probability of flipping heads on a coin three times in a row?

P(H, H, H) = 8

1

2

1

2

1

2

1

2704

16

52

4

52

4

Example 3:A die is rolled twice. What’s the probability of rolling a 2 and then an even number?

Solution:

Example 4:You spin the spinner 3 times. What is the probability of spinning a 4, a 3 and then a 1?

Solution:

12

1

2

1

6

1

512

6

8

3

8

2

8

1

256

3

Dependent: Two events such that the occurrence of one affects the occurrence of the other.P(A and B) = P(A) P(B|A)

**P(B|A) = means the probability of B given that event A has already occurred.

Example 1:What is the probability of drawing a King and then an Ace without replacement?

P(King, Ace) =

265216

514

524

Example 2:You randomly select two marbles from a bag that contains 14 green, 7 blue, and 9 red marbles. What is the probability that the first marble is blue and the second marble is not blue if you do not replace the first marble?

Solution:870

161

29

23

30

7

Example 3:Your teacher passes around a basket with 6 red erasers, 9 blue erasers, and 7 green erasers. If you and your two neighbors are the first to randomly select an eraser, what is the probability that all three of you select green erasers?

Solution:P(A) and P(B|A) and P(C|A and B) 9240

210

20

5

21

6

22

7

The table shows the number of males and females with certain hair colors. Find …

A) the probability that a listed person has red hair

B) the probability that a female has red hair

Brown Brown hairhair

Blonde Blonde hairhair

Red Red hairhair

Black Black hairhair

OtherOther

MaleMale 4242 1111 33 1717 2727

FemaleFemale 4747 1616 1313 99 1515

P(red hair) = # of people with red hair total # of people

P(red hair | female) = # of red hair females

total # of females

25

2

200

16

100

13

When you consider the outcomes for either of two events A and B, you form the union of A and B.

When you consider only the outcomes shared by both A and B, you form the intersection of A and B.

When the sets of A and B have nothing in common (no intersection) then they are considered mutually exclusive events.

If A and B are mutually exclusive events (one event does not have anything in common with the other), then…

P(A or B) = P(A) + P(B)

A die is rolled one time. What is the probability of rolling a 2 or a 6?

Solution:P(A or B) = P(A) + P(B) =

6

1

6

1

6

2

3

1

A card is randomly selected out of a standard deck of 52 cards. What is the probability that it is a 2 or a king?

P(A or B) = P(A) + P(B) =

52

4

52

4

52

8

13

2

If A and B are not mutually exclusive, then there are some outcomes in common.

Therefore, the intersection of A and B are counted twice when P(A) and P(B) are added.

So, P(A and B) must be subtracted once from the sum…

P(A or B) = P(A) + P(B) – P(A and B)

A die is rolled one time. What is the probability of rolling an odd number or a prime number?

Odd = 1, 3, 5 P(A) = Prime = 2, 3, 5 P(B) =

Odd and prime = 3, 5 P(A and B) =

6

3

6

3

6

2

A card is randomly selected from standard deck of 52 cards. What is the probability that it is a red card or a king?

Red cards = 26 =Kings = 4 =

Red Kings = 2 =

52

26

52

4

52

2

The probability that it will rain today is 40%. The probability that is will rain tomorrow is 30%. The probability that it will rain both days is 20%. Find the probability that it will rain either today OR tomorrow.

Solution: P(A) + P(B) – P(A and B)

P(today) + P(tomorrow) – P(today and tomorrow) = %50%20%30%40

The expected value is often referred to as the “long-term” average or mean .

This means that over the long term of doing an experiment over and over, you would expect this average.

To find the expected value or long term average, simply multiply each value of the random variable by its probability and add the products.

Suppose that the following game is played. A man rolls a die. If he rolls a 1, 3, or 5, he loses $3, if he rolls a 4 or 6, he loses $2, and if he rolls a 2, he wins $12. What gains or losses should he expect on average? (What is his expected value?)

We must find the probabilities of each outcome. We can make a chart to help us see this.

Possible Losses/Gains

Probability P(losses/gains)

-$3

-$2

$12

6

3

6

2

6

1

Now, we multiply the probability for each outcome by the amount of money either gained or lost for that outcome.

Expected value of rolling a 1, 3, or 5: ________________

Expected value of rolling a 4 or 6: ________________

Expected value of rolling a 2: _______________

50.1$6

9

6

33$

67.0$6

4

6

22$

00.2$6

12

6

112$

To find the expected value for the entire game (the answer), simply add up the expected value for each outcome.

Expected Value =

This means that the man playing this game is expected to lose an average of $0.17 each game he plays.

17.0$00.2$67.0$50.1$