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# 2.12 13.08 Counting Principles

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### 2.12 13.08 Counting Principles

1. 1. Disjoint Events A and B are disjoint.
2. 2. Disjoint Events A and B are not disjoint.
3. 3. Probability of the Union of Events A B
4. 4. Complementary Events A NotA
5. 5. If two fair dice are tossed, what is the probability that the sum is a 2 or a 3?
6. 6. Thirteen of the 50 states include territory that lies west of the continental divide. Forty-two states include territory that lies east of the continental divide. Is this possible? What is the continental divide?
7. 7. Three fair coins are tossed. What is the probability that not all the coins show the same face?
8. 8. Multiplication Counting Principles Then number of ways to choose one element from A and one element from B is N(A)*N(B). For example, if: N(A)=6 N(B)=8 If you choose one member of A and one member of B, the list of total choices is: There are 48 possible choices.
9. 9. Tree Diagrams Sometimes it is useful to draw a Tree Diagram. For example, if you are choosing between two majors at 5 colleges, and deciding whether or not to play a sport: Haverford Hamilton SJU Conn. Coll. NYU Haverford Hamilton SJU Conn. Coll. NYU Engineering Math Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport Sport No Sport
10. 10. Selection with Replacement If you choose a numbered ping pong ball from a box 3 times, and there are 3 ping pong balls, how many possible combinations are there? Selection without Replacement 1 2 3 1 2 3 1 2 3 1 2 1 3 x 3 x 3 = 27 Choices 3 x 2 x 1 =6 Choices 1 2 3
11. 11. A quiz has 10 questions, each of which can be answered “always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly? Since there is only one correct combination, the probability of choosing that combination is 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices 3 Choices
12. 12. How many license plates are possible with 4 letters from A to Z (except O and I) followed by two digits from 0 through 9? 10 Choices 10 Choices 24 Choices 24 Choices 24 Choices 24 Choices