1. Will it probably happen or not?
Probability
Will it probably happen or not?

2. Probability is a measure of how likely it is for an event to happen.
We name a probability with a number from 0 to 1.
If an event is certain to happen, then the probability of the event is 1.
If an event is certain not to happen, then the probability of the event is 0.

3. If it is uncertain whether or not an event will happen, then its probability is some fraction between 0 and 1 (or a fraction converted to a percent).

4. Overview You can represent the probability of an event by
marking it on a number line like this one
Impossible
0 = 0%
50 – 50 Chance
½ , .5, 50%
Certain
1 = 100%
The language of probability includes:
Experiment – a systematic investigation where the answer is unknown
Trial – one specific instance of an experiment
Outcome - the result of a single trial
Event – a selected outcome, such as getting an 11 from rolling two dice
Event Space/or Sample Space – the set of all possible outcomes of an experiment

5. Probability Definition. The likelihood of an event.
Expressed as a fraction or percent.
There is a 30% chance of rain tomorrow.
She has a 40% chance of scoring in a game.

6. Basic concepts To determine the probability we have two approaches.
Look at the past to predict the future (analyze data). This is called EXPERIMENTAL probability.
Look at all the possibilities. This is called THEORETICAL probability.

7. The probability of an event is
P = number of favorable events
total number of outcomes

8. Basic Concepts
We write probabilities as a ratio--these ratios can then be written as fractions or percents.
0 means that the probability of something happening is impossible.
1 means that the probability of something happening is certain.

10. THEORETICAL PROBABILITY
We said that probability was the ratio of the number of favorable outcomes of an experiment to the total number of possible outcomes.
What happens if we don’t do the experiment?
This is called THEORETICAL PROBABILITY.
We look at all the POSSIBLE FAVORABLE outcomes and compare it with the possible outcomes.

11. A B C D
1. What is the probability that the spinner will stop on part A?
What is the probability that the spinner will stop on
An even number?
An odd number? 3 1 2
3. What fraction names the probability that the spinner will stop in the area marked A? A C B

12. Lawrence is the captain of his track team
Lawrence is the captain of his track team. The team is deciding on a color and all eight members wrote their choice down on equal size cards. If Lawrence picks one card at random, what is the probability that he will pick Blue?
yellow
green
black
red
blue
black
blue
blue

17. Complementary Events
The probability of something NOT happening is 1 – P(event happening)
Probability of NOT rolling a 5 is 1 – 1/6 = 5/6

18. Example
There are two sports of a particular high school (mostly girls school): basketball and soccer. The number of students enrolled in each program is given in the table on the next slide. The row total gives the total number of each category and the number in the bottom-right cell gives the total number of students. A single student is selected at random from this high school. Assuming that each student is equally likely to be chosen, find :
Barnett/Ziegler/Byleen Finite Mathematics 11e

19. 3. P(boy who plays basketball) 4. P(boy who plays soccer)
girls
boys
Total
Basketball 53 47
100
Soccer 37 13 50 90 60
150
1. P(basketball)
2. P(boy)
3. P(boy who plays basketball)
4. P(boy who plays soccer)
= 100/150 = 2/3
= 60/150 = 2/5
= 47/150
= 13/150
Barnett/Ziegler/Byleen Finite Mathematics 11e

20. Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
Girls
Boys
Total
Basketball 53 47
100
Soccer 37 13 50 90 60
150
Given that a girl is selected at random, what is the probability that this student is a basketball player?
Restricting our attention to the column representing undergrads, we find that of the 90 girls, 53 are basketball players. Therefore, P(G|S)=53/90
Barnett/Ziegler/Byleen Finite Mathematics 11e

21. Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
Girls
Boys
Total
Basketball 53 47
100
Soccer 37 13 50 90 60
150
Given that an soccer player is selected, find the probability that the student is an girl.
Restricting the sample space to the 50 soccer players, 37 of the 50 are girls. Therefore, P(G|S) = 37/50 = 0.74.
Barnett/Ziegler/Byleen Finite Mathematics 11e

22. I have a brown eyed mother and a blue eyed dad
I have a brown eyed mother and a blue eyed dad. What is the probability that they will have a blue eyed baby?

23. My Mother
My mother has brown eyes. Her genes are B and b. Her genotype is Bb.
Her B gene is the dominant brown eye trait. Her b gene is the recessive blue eye trait.
Since she has one dominant and one recessive allele she has the dominant brown eye trait.

24. My Father
My father has blue eyes. His genes are b and b. His genotype is bb.
His b genes are both recessive.
Since he has two recessive genes he has the recessive blue eye trait.

25. So What About Me?
I have one gene from my mother and one gene from my father.
My genotype is determined by these genes.
I will have brown eyes if my genotype is BB or Bb.
I will have blue eyes if my genotype is bb.

26. Now Using a Tree Diagram!
Mother’s Genotype
Father’s Genotype
Sample Space B b Bb
I have a 50% chance of having blue eyes. b Bb b b bb b bb

27. Two events Problem – this takes up a bunch of paper.
One way to determine the probability of two events (both parents giving a blue-eye gene) is to draw a tree diagram.
You enter an ice cream shop. They have vanilla, chocolate and strawberry ice cream. They have sprinkles and nuts as toppings.
What is the probability that you will get a vanilla cone with sprinkles if you randomly pick a cone?
Problem – this takes up a bunch of paper.
There is an easier way.

28. Two events
To find the probability of two events, you can multiply the probability of the the first and the probability of the
second.
P(blue eyes) =
P(vanilla and sprinkles) =

29. Independent vs. Dependent
With multiple events, when the second outcome IS NOT related to the first outcome, we say they are independent.
With two or more events, when the second outcome IS related to the first outcome, we say they are dependent.

30. There are a few different situations I could encounter.
Since the probability is the ratio of the desired outcomes to the total outcomes, I need to know how to count outcomes.
There are a few different situations I could encounter.
One event (pretty easy - look at the possibilities)
More than one event
Are they independent or dependent?
Are they mutually exclusive
How do I count them?

31. One event On a certain die, there are 3 fours, 2 fives, and 1 six.
P(rolling an odd) =
P(rolling a number less than 6) =
P(rolling a 6) =
P(not rolling a 6) =
P(rolling a 2) =
1/3
5/6
1/6
5/6

32. Two events
I have 6 blue marbles and 4 red marbles in a bag. If I do not replace the marbles, …
P(blue) =
P(red) =
P(blue, blue) =
P(red, blue) =
P(blue, red) =
Is this an example of independent or dependent events?
dependent

33. Two events
There are 8 girls and 7 boys in my class, who want to be line leader or lunch helper, …
P(G: LL, B: LH) =
P(G: LL, G: LH) =
P(B: LL, B: LH) =

34. Watch the wording… Suppose I flip a coin. P(H) = P(T) = P(H or T) =
P(H and T) =
1/2
1/2 1

35. True/False
Suppose you have a true/false section on Wednesday’s test. If there are 4 questions,…
P(all 4 are true) =
P(all 4 are false) =
Is this an example of independent or dependent events?
Let’s see if guessing is good.

36. Fundamental Counting Principle
So, for the true/false scenario, it would be: true or false for each question.
Therefore you have a chance of getting one true.
So you have a x x x or a chance of all being true.
So P(all 4 are true) =
What is P(all four are false)?

37. Fundamental Counting Principle
Suppose you have 5 multiple-choice problems, each with 4 choices. How many different ways can you answer these problems?
4 • 4 • 4 • 4 • 4 = 1024
What is the probability that you will guess and get all 5 correct?

38. Fundamental Counting Principle
Now, suppose the question is matching: there are 6 questions and 10 possible choices. Now, how many ways can you match?
• • • • • =
151,200 7 6 5 10 9 8

39. How are true/false and multiple choice questions different from matching questions?

40. In a shipment of 20 computers, 3 are defective
In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested?
x x =

41. I can use this the other way.
Are smoking and lung disease related?
Smoker
Non-
smoker
Has
Lung
Disease
0.12
0.03
No Lung
0.19
0.66
If the two events are independent then P(smoker) x P(lung disease) = P(smoker and lung disease).
Barnett/Ziegler/Byleen Finite Mathematics 11e

42. Another Example of Dependent Events
Are smoking and lung disease related?
Step 1. Find the probability of lung disease.
P(L) = 0.15 (row total)
Step 2. Find the probability of being a smoker
P(S) = 0.31 (column total)
Step 3. Check
0.15 x 0.31 = 0.465
P(L and S) = 0.12  0.465
L and S are dependent.
Smoker
Non-
smoker
Has
Lung
Disease
0.12
0.03
No Lung
0.19
0.66
Barnett/Ziegler/Byleen Finite Mathematics 11e

43. At a fast food restaurant, 90% of the customers order a burger
At a fast food restaurant, 90% of the customers order a burger. If 72% of the customers order a burger and fries, what is the probability that a customer who orders a burger will also order fries?
Burger
No burger
total
Fries 72
100
No fries 18 90 10

44. Barnett/Ziegler/Byleen Finite Mathematics 11e
Another Example
Two machines are in operation.  Machine A produces 60% of the items, whereas machine B produces the remaining 40%.  Machine A produces 4% defective items whereas machine B produces 5% defective items.  An item is chosen at random.
What is the probability that it is defective?
0.04 def A
0.96 good
60%
40%
0.05 def B
0.95 good
Barnett/Ziegler/Byleen Finite Mathematics 11e