1. Chapter 5- Probability Review

2. Section 5.1 An event is the set of possible outcomes
Probability is between 0 and 1
The event A has a complement, the event not A. Together these two probabilities sum 1.
ex. At least one and none are complements
Probability of an event =
number of outcomes in event / number of equally likely outcomes

3. Section 5.1
Probability Distributions give all values resulting from a random process.
Sample space is the complete list of disjoint outcomes. All outcomes in a sample space must have total probability equal to 1.
Ex. The sample space for rolling a die is {1,2,3,4,5,6} The sample space for rolling two die is the table of 36 outcomes we’ve seen.

4. Disjoint Disjoint and mutually exclusive mean the same thing.
Disjoint means two different outcomes can’t occur on the same opportunity
Ex. Can’t roll an extra credit and no collect on the same roll. Can’t get a heads and tail on the same flip.
These items on a Venn diagram would have no intersection

5. Disjoint Continued
Flipping a coin and getting a head and tail is disjoint
Flipping a coin twice and getting a head and tail is disjoint.

6. Law of Large Numbers
In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the population proportion.
The difference between a sample proportion and the population proportion must get smaller as the sample size gets larger.

7. Fundamental Principle of Counting
Processes can be split into stages (Flip coin once, then again)
If there are k stages with different possible outcomes for each stage, the number of total possible outcomes is
n1*n2*n3*n4… nk
Ex. How many total outcome for rolling a die and flipping a coin?
6*2 = 12

8. Probability Simulations
Assumptions
What probability are you assuming? Are events independent?

9. Probability Simulations
2) Model *How specifically will you use your table of random digits? *Make sure to say what to do with repeats, unallocated numbers *Fully describe what constitutes a run and what statistics you’re collecting. *Make a table to show how digits /groups are assigned

10. Probability Simulations
3) Repetition Run the simulations and record the results in a frequency table.

11. Probability Simulations
4) Conclusion Write the conclusion in context of the situation. Be sure to say the probability is ESTIMATED.

12. Addition Rule of Probability
Remember first that “or” means one or the other or both
P(A or B) = P(A) + P(B) – P(A and B)
If A and B are disjoint, there is no intersection. Therefore, P(A and B)= 0.
If A and B are disjoint: P(A or B) = P(A) + P(B)

13. Conditional Probability and the Multiplication Rule
The probability of an event A and B both happening is
P(A and B) = P(A) * P(B|A)
P(A and B) = P(B) * P(A|B)
The probability of an event changes based on what happened before.

14. Conditional Probability
Rearranging means
P(A|B) = P(A and B) / P(B)
Probability of both divided by the probability of the first event.

15. Independent Events
The occurrence of one event doesn’t change the probability of the second event occurring
Test for Independence:
IS P(A|B) = P(A) or P(B|A) = P(B) ?
If yes, you have independent events.
Sometimes this isn’t obvious that one has an effect without checking

16. Multiplication Rule for Independent Events
Remember if events are independent,
P(A|B) = P(A) or P(B|A) = P(B)
Therefore, since P(A and B) = P(A) * P(B|A)
P(A and B) = P(A) * P(B)
for independent events

17. Review Questions
If events A and B are independent and P(A) = 0.3 and P(B) = 0.5, then which of these is true? A. P(A and B) = 0.8 B. P(A or B) = 0.15 C. P(A or B) = 0.8 D. P(A | B) = 0.3 E. P(A | B) = 0.5

18. Answer
If events A and B are independent and P(A) = 0.3 and P(B) = 0.5, then which of these is true?
A. P(A and B) = 0.8
B. P(A or B) = 0.15
C. P(A or B) = 0.8
P(A | B) = 0.3
Probability of A is not changed based on the occurrence of event B
E. P(A | B) = 0.5

19. Two Way Table Questions
Crash Type
Single Vehicle
Multiple Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol Related
11,345
11,336
22,681
22,086
16,223
38,309
If a fatal auto crash is chosen at random, what is the approximate probability that the crash was alcohol related, given that it involved a single vehicle?
A. 0.28
B. 0.49
C. 0.58
D. 0.69
E. The answer cannot be determined from the information given.

20. Answer 0.49 10,741/22,086 = 0.49 Crash Type Single Vehicle
Multiple Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol Related
11,345
11,336
22,681
22,086
16,223
38,309
If a fatal auto crash is chosen at random, what is the approximate probability that the crash was alcohol related, given that it involved a single vehicle?
0.49
10,741/22,086 = 0.49

21. Two Way Table Questions
Crash Type
Single Vehicle
Multiple Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol Related
11,345
11,336
22,681
22,086
16,223
38,309
What is the approximate probability that a randomly chosen fatal auto crash involves a single vehicle and is alcohol related?
A. 0.28
B. 0.49
C. 0.58
D. 0.69
E. The answer cannot be determined from the information given

22. Answer 0.28 Because 10,741/38,309 = 0.28 Crash Type Single Vehicle
Multiple Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol Related
11,345
11,336
22,681
22,086
16,223
38,309
What is the approximate probability that a randomly chosen fatal auto crash involves a single vehicle and is alcohol related?
0.28
Because 10,741/38,309 = 0.28

23. Review Question
For all events A and B, P(A and B) = A. P(A) · P(B) B. P(B | A) C. P(A | B) D. P(A) + P(B) E. P(B) · P(A | B)

24. Answer P(B) · P(A | B) This is the multiplication rule.
For all events A and B, P(A and B) =
A. P(A) · P(B)
B. P(B | A)
C. P(A | B)
D. P(A) + P(B)
P(B) · P(A | B)
This is the multiplication rule.
If they are independent,
P(A|B) = P(A)

25. Review Question
The mathematics department at a school has twenty instructors. Six are easy graders. Twelve are considered to be good teachers. Seven are neither. If a student is assigned randomly to one of the easy graders, what is the probability that the instructor will also be good? A. 7/20 B. 5/12 C. 7/12 D. 5/6 E. The answer cannot be determined from the information given.

26. Answer
D. 5/6
Easy
Not Easy
Total
Good 5 7 12
Not Good 1 8 6 14 20

27. Review Question
The management of Young & Sons Sporting Supply, Inc., is responding to a claim of discrimination. If the company has employed the 60 people in this table, how many females over the age of 40 must the company hire so that the age and sex of its employees are independent?
40 Years
> 40 Years
Total
Male 25 15
Female 20

28. Answer
First, let N be the number of females older than 40 to be hired.
Set up a proportion so
P(male |<40) = P(Female|< 40)
N=12
40 Years
> 40 Years
Total
Male 25 15 40
Female 20 N
20+N 45
15+N
60+N

29. Review Question
If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.08, which of these is true?
Events A and B are independent and mutually exclusive.
Events A and B are independent but not mutually exclusive.
Events A and B are mutually exclusive but not independent.
Events A and B are neither independent nor mutually exclusive.
Events A and B are independent, but whether A and B are mutually exclusive cannot be determined from the given information.

30. Answer Events A and B are independent but not mutually exclusive.
If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.08, which of these is true?
Events A and B are independent and mutually exclusive.
Events A and B are independent but not mutually exclusive.
Events A and B are mutually exclusive but not independent.
Events A and B are neither independent nor mutually exclusive.
Events A and B are independent, but whether A and B are mutually exclusive cannot be determined from the given information.