Union, Intersection

Intersection of Sets

3

Intersection of Sets

• Intersection (∩) [of 2 sets]: the elements common to both sets

• Some guidelines when finding the intersection of 2 sets:– Usually easier to start with the set containing

the least number of elements

4

Solving Compound Inequalities Using Intersection

• Can be found in two formats:– Two linear inequalities separated by the word and

– A statement containing two inequality symbols• -4 < x < 7

Union of Sets

6

Union of Sets

• Union (U) [of 2 sets]: the combination of the distinct elements from both sets

• Some guidelines when finding the union of 2 sets:– “Dump” the elements of both sets together

and remove the duplicates

1

2

1

51 1 1

2 5 10

S

T

R

O

P1

2

3

6

5

4

Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel?

P(even) = (3 evens out of 6 outcomes)

(1 vowel out of 5 outcomes)P(vowel) =

P(even, vowel) =

Independent Events

Slide 7

Dependent Event

• What happens the during the second event depends upon what happened before.

• In other words, the result of the second event will change because of what happened first.

The probability of two dependent events, A and B, is equal to the probability of event A times the probability of event B. However, the probability of event B now depends on event A.

P(A, B) = P(A) P(B)Slide 8

Dependent Event

6 3 or

14 7

5

13

3 5 15 or

7 13 91

Example: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first, what is the probability that you will get 2 black pens?

P(black second) = (There are 13 pens left and 5 are black)

P(black first) =

P(black, black) =

THEREFORE………………………………………………

Slide 9

TEST YOURSELFAre these dependent or independent events?

1. Tossing two dice and getting a 6 on both of them.

2. You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose one marble out of the bag, look at it then put it back. Then you choose another marble.

3. You have a basket of socks. You need to find the probability of pulling out a black sock and its matching black sock without putting the first sock back.

4. You pick the letter Q from a bag containing all the letters of the alphabet. You do not put the Q back in the bag before you pick another tile.

Slide 10

Find the probability

• P(jack, factor of 12) 1

5

5

8x =

5

40

1

8

Independent Events

Slide 11

Find the probability

• P(6, not 5)1

6

5

6x =

5

36

Independent Events

Slide 12

Find the probability• P(Q, Q)• All the letters of the

alphabet are in the bag 1 time

• Do not replace the letter

1

26

0

25x =

0

650

0

Dependent Events

Slide 13