1. 12.1 The Counting Principle (Crash Course in Probability)

2. Probability
The risk of getting struck by lightning in any year is 1 in 750,000. The chances of surviving a lightning strike are 3 in 4. These risks and chances are a way of describing the probability of an event.
The probability of an event is a ratio that measures the chances of the event occur.

3. Introduction to Probability
The Probability of an event occurring is a number between 0 and 1.
0 means that the event cannot occur…
1 means that the event is certain to occur.
The probability that event A will occur is
P(A) = number of outcomes in A total number of outcomes

4. P(3)= P(prime #)= P(factors of 8)= P(9)= 1 2 3 4 5 6 7 8
Quick Examples of Probability: 1 2 3 4 5 6 7 8
P(3)=
P(prime #)=
P(factors of 8)=
P(9)=

5. How can you count the maximum number of license plates a state can issue?
It could take a long time to list all of the possible plate numbers We can use counting principles to save time

6. Outcome: the result of a single trial
Vocabulary:
Outcome: the result of a single trial
Ex. Flipping a coin has two outcomes: heads or tails
Sample Space: the set of all possible outcomes
Ex. The sample space of the courses offered next year is Pre-Calc, College Algebra/Trig and statistics.

7. Event: consists of one or more outcomes of a trial
Ex. The event was the students taking the test; while the sample space is the range of scores from

8. Independent events: when each outcome does not affect the others within the event.
Ex. Having twenty students choose a number from is an independent event.
Ex. Having twenty students choose a number from a hat where there are one hundred cards; labeled with each number from 1-100; is a dependent event.

9. A multiple choice test has 6 questions with 5 choices each.
In how many ways can you complete the test?

10. How many possible 7-digit phone numbers are there if the
first digit cannot be 0 or 1?

11. Objective -To count the number of ways an
event can happen.

12. Independent or dependent?
Example 1.
A pizza place offers customers a choice of American, mozzarella, Swiss, feta, or provolone cheese with one topping chosen from pepperoni, mushrooms or sausage. How many different combinations of cheese and toppings are there?
Independent or dependent?
(5)(3)
= 15
Cheese
meat
Example 1-1b

13. Multiple-Choice Test Item
For their vacation, the Murray family is choosing a trip to the beach or to the mountains. They can select their transportation from a car, plane, or train. How many different ways can they select a destination followed by a means of transportation?
A B C D 9
Independent or dependent?
Their choice of destination does not affect their choice of transportation, so these events are independent.
Example 1-2a

14. Solve the Test Item
There are two ways they can choose a destination and 3 ways they can choose a means of transportation. By the Fundamental Counting Principle, there are 2 • 3 or 6 total ways to choose.
A B C D 9
Answer: C
Example 1-2a

15. Answer: So, there are or 100 possible different codes.
Communication How many codes are possible if an answering machine requires a 2-digit code to retrieve messages?
The choice of any digit does not affect the other digit, so the choices of digits are independent events.
There are 10 possible choices for the first digit and 10 possible choices for the second digit.
Answer: So, there are or 100 possible different codes.
Example 1-3a

16. Dependent events: the outcome of one event does affect the outcome of another event.
The fundamental counting principle applies to dependent events as well as independent events.
Ex. Taking a yellow skittle from a jar and then taking a second skittle without replacing the yellow one; is a dependent event.
Often time factorials can be used to represent a dependent event. (recall; )

17. Factorial on the calculator
math
PRB
Fourth option down

18. How many different schedules could a student have who is planning to take 4 different classes? Assume each class is offered each period.
When a student schedules a given class for a given period, he or she cannot schedule that class for any other period. Therefore, the choices of which class to schedule each period are dependent events.
There are 4 classes that can be taken during the first period. That leaves 3 classes for the second period, 2 classes for the third period, and so on.
Example 1-4a

19. Period 1st 2nd 3rd 4th Number of Choices 4 3 2 1
Answer: There are or 24 different schedules for a student who is taking 4 classes.
Example 1-4a

20. Ex5. The number of ways to organize your 7 books on a shelf
Ex5. The number of ways to organize your 7 books on a shelf. However, the math textbook must be first for easy reference.
Dependent event

21. Ex6. A certain state has license plates that are made of four numbers followed by two letters? How many different license plates are possible if no repeated letters or numbers are allowed?

22. Find the number of permutations of 7 objects taken 4 at a time.

23. Find the number of permutations for the letters in the
word STREET.

24. Find the number of permutations for the letters in the
word MATHEMATICS.
Find the number of permutations for the letters in the
word SLEEPLESS.