Probability & Expected Value

Math I

MM1D2:Students will use the basic laws of probability

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

Simple Probability…(review)

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

Independent Events

Examples: 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

121

21

21

270416

524

524

More Examples… 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:

121

21

61

5126

83

82

81

2563

MMD12b:Find the probabilities of dependent events

Dependent Events: 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.

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

P(King, Ace) =

265216

514

524

265216

514

524

More Examples… 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:870161

2923

307

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)

9240210

205

216

227

9240210

205

216

227

Probability from a Table: 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 hair

Blonde hair

Red hair

Black hair

Other

Male 42 11 3 17 27

Female 47 16 13 9 15

Solution: P(red hair) = # of people with red hair

total # of people

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

total # of females

252

20016

252

20016

10013

Probability of Compound Events

Compound Events: UnionWhen you consider the outcomes for either of two events A and B, you form the union of A and B.

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

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

Probability of Mutually Exclusive Compound 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)

Example 1: 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

161

62

31

Example 2: 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) =

524

524

528

132

Probability of Compound Events NOT

Mutually Exclusive: 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)

Example 3: 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) =

63

63

63

62

Example 4: 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 =

5226

524

522

Extension: 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