1. How can the likelihood of an event be measured?
Chapter 6 Probability
How can the likelihood of an event be measured?

2. Chance Experiments and Events 6-1
What is a chance experiment and how can those chances be represented?

3. Chance Experiment—any activity where the outcome has two or more possibilities
Sample Space—the set of all possible outcomes
Tree Diagram—one method of finding the sample space that shows/details all the outcomes

4. example Show all possible outcome for the situation:
a woman and a man walk into a sport equipment store. Each one could buy a tennis racket, a soccer ball or a Frisbee.
Tennis racket
Soccer ball
man
Frisbee
woman
Tennis racket
Soccer ball
Frisbee

5. Simple event—consists of exactly one outcome from the sample space
Set notation—
(event1, event2, )
Event—any collection of outcomes from the sample space of a chance experiment
Simple event—consists of exactly one outcome from the sample space

6. The complement of an event A—all the outcomes that are not in A
Noted as A’, Ac , or
A or B—something which is in the sample space of A or the sample space of B
A U B (A union B)
A and B—something which is in the intersection of the sample space of A and of B
(A intersect B)
Disjoint or mutually exclusive events—two events that have no common outcomes

7. Venn Diagrams A or B A and B Disjoint Events A’
Shaded area represents the indicated statement
A or B
A and B
Disjoint Events A’

8. Homework Pg , 4, 7, 10, 12

9. The definition of Probability 6-2
What is probability?

10. Probability Called the classical approach and works well when you have a finite number of outcomes

11. Law of Large Numbers (James Bernoulli)
As the number of repetitions of a chance experiment increases the chance that the relative frequency of an event will differ from the true probability of the event by more than a small number approaches zero
In English:
The more trials you do for an experiment, the closer the rel. freq. you get will be to the true probability.

12. Relative Frequency Approach

13. Subjective Approach
Personally assigned values which are not replicable
not as much value is placed on these

14. Basic properties of probability 6-3
What are the basic properties of probability?

15. For any event E 0≤p(E)≤1
If S is the sample space for an experiment then P(S) = 1
If two events are disjoint, P(E or F) = P(E) +P(F)
For any event E
P(E) + P(not E) = 1
P(E) =1 – P(not E) or P(not E) = 1- P(E)

16. Homework P evens

17. Conditional probability 6-4
What is conditional probability and how can it be found?

18. Conditional Probability occurs when the likelihood that one event will occur is changed by what has happened previously.

19. Example
There is a disease that effects .1% of the population. A test to determine if you have the disease exists. The test can give a false positive 20% of the time. IF
E= the individual has the disease
F= the individual tested positive for the disease
What is
P(E)=.001=.1%
P(E|F)=.8=80%
The probability of E given that F occurred
The occurrence of E is unlikely in the general population however knowing that the test is positive it is very likely that you have the disease.

20. Conditional Probability Formula: Read the probability of E given F Hint: intersection = multiplication union = addition

21. Example
Consider the population to be all families with two children. Assuming that there is an equal chance of having a boy or a girl there are 4 possible outcomes. GG, BB, GB, BG (ordered by age)
1) What is the probability of choosing a family with two girls given the family has at least one girl?
2) What is the probability that they have 2 girls given that the oldest is a girl?

22. Example
In 1912, the titanic sank. Almost 1500 people died, most of them men. Was that because a man was less likely to survive than a woman or because men outnumbered women 3 to 1? Titanic Survival Data:
Female Survival Rate p(S|F)
Male Survival Rate P(S|M)
Overall Survival Rate P(S|on the ship)
Male
Female
Totals
Survived
367
344
711
Died
1364
126
1490
totals
1731
470
2201
Lore is likely to be correct. Women and children first!

23. Homework pg even

24. Independence
What are independent events and how are their probabilities found?

25. P(not E|not F) = P(not E)
Independent events have no bearing on each other even if you know the outcome of the 1st event
P(E|F) = P(E)
this also implies
P(not E|F) = P(not E)
P(E| not F) = P(E)
P(not E|not F) = P(not E)
Basically logical

26. Probability of 2 independent events occurring follows from:
Cross multiply:
But if E and F are Independent P(E|F) = P(E) we substitute to find

27. Replacement—not removing a selection from the population (look at it and put it back).
What is the probability of getting a red marble on the second try given that you had 5 red and 5green marbles in the bag and replaced the first marble after choosing it.
Without Replacement—removing the selection from the population once it is selected.
What is the probability of getting a red marble on the second try given that you had 5 red and 5green marbles in the bag and did not replace the first marble after choosing it.

28. Homework Pg 267-270 36 – 52 even and Pg 259 #33

29. Some General probability rules 6-6
What are some basic or general probability rules?

30. THE UNION OF EVENTS: P(A U B) = P(A) + P(B) – P(A ∩ B) Inclusive—events that share some solutions 0≤P(A∩B)≤1 Mutually exclusive events do not have any overlap
P(A∩B)=0

31. REMEMBER: P(E∩F)=P(E|F)P(F) if they are independent but =P(E)P(F)
Law of Total Probability
if B1, B2, …, Bn are disjoint events with
P(E∩B1) + P(E∩B2) + … + P(E∩Bn) = 1
then
P(E) = 1
P(E) = P(E|B1)P(B1) +P(E|B2)P(B2) + … + P(E|Bn)P(Bn)

32. Baye’s Rule
if the items are disjoint and you want the probability of P(B1|E) but have P(E|B1)
Knowing
And Or
Then by substitution:
P(B1|E) =
P(B1)P(E|B1)
P(E|B1)P(B1) +P(E|B2)P(B2) + … + P(E|Bn)P(Bn)

33. S1 = service 1 S2 = service 2 L = late
Example pg 279 #6.27
Two shipping services offer overnight delivery of parcels, and both promise delivery before 10AM. A mail-order catalog company ships 30% of its overnight packages using shipping service 1 and 70% using service 2. Service 1 fails to meet the 10AM delivery promise 10% of the time, whereas service 2 fails to deliver by 10 AM 8% of the time. Suppose you made a purchase from this company and were expecting your package by 10 AM but it is late. Which service is more likely to have been used—ie who should you call to see where your important package is?
S1 = service S2 = service L = late
We know P(S1) = P(S2) = P(L|S1) = P(L|S2) = .08
Since you know the package is late you need to find P(S1|L) and P(S2|L)
So you should call service two to see where the package is because even though they have a lower overall percentage rate (.08), they have 70% of the company’s business so 65% of the late packages from this company are from service 2.
(it’s a lot of math this time, but if you are constantly having to call it would be beneficial to know who to call first and save time overall by only making 1 call 65% of the time)

34. Homework pg even

35. Estimating probabilities empirically and using simulations 6-7
How do you accurately estimate probabilities?

36. Simulation provides a method of estimating probabilities that we do not have the time or resources to determine analytically and are not practical to observe or estimate empirically. Simulation generates “observations” by performing an experiment that is as similar as possible in structure to the real situation.

37. Example:
A professor wants to simulate the possible scores on a 20 question true/false quiz when the students are guessing. Rather than finding 500 students to write down 20 answers and then scoring each paper.
You could place 20 balls 10 red/10 blue in a box, allow blue to represent true, make a selection, replace the ball and repeat 20 times “per quiz”, record the # of corrects. Repeat 1000 times and construct a table of probability.
Why 20 balls with 10 of each?
to make it random but equally likely to pick either color OR
You could use a random number table allowing evens to be correct and odds incorrect break the table into groups of 20 and count the number of corrects in each.

38. Homework pg 291 – 294 even

39. Review Pg – 297 even and 81