1. Probability II

2. Probability Denoted by P(Event)
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely.

3. Experimental Probability
The relative frequency at which a chance experiment occurs
Flip a fair coin 30 times & get 17 heads

4. Law of Large Numbers
As the number of repetitions of a chance experiment increase, the difference between the relative frequency of occurrence for an event and the true probability approaches zero.

5. Basic Rules of Probability
Rule 1. Legitimate Values
For any event E,
0 < P(E) < 1
Rule 2. Sample space
If S is the sample space,
P(S) = 1

6. Rule 3. Complement
For any event E,
P(E) + P(not E) = 1

7. Rule 4. Addition
If two events E & F are disjoint,
P(E or F) = P(E) + P(F)
(General) If two events E & F are not disjoint,
P(E or F) = P(E) + P(F) – P(E & F)

8. Are these disjoint events?
Ex 1) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. (Note: these are not simple events since there are many types of each brand.) Suppose that P(H) = .25, P(N) = .18, P(T) = .14.
Are these disjoint events?
yes
P(H or N or T) =
= .57
P(not (H or N or T) =
= .43

9. Ex. 2) Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is What is the probability that they like country or jazz?
P(C or J) = = .6

10. Independent Independent Not independent
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs
A randomly selected student is female - What is the probability she plays soccer for PWSH?
A randomly selected student is female - What is the probability she plays football for PWSH?
Independent
Not independent

11. Rule 5. Multiplication
If two events A & B are independent,
General rule:

12. Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that both bulbs are defective?
Can you assume they are independent?

13. Disjoint events are dependent!
Ex 4)
If P(A) = 0.45, P(B) = 0.35, and A & B are independent, find P(A or B).
Is A & B disjoint?
NO, independent events cannot be disjoint
If A & B are disjoint, are they independent?
Disjoint events do not happen at the same time.
So, if A occurs, can B occur?
Disjoint events are dependent!
P(A or B) = P(A) + P(B) – P(A & B)
If independent, multiply
How can you find the probability of A & B?
P(A or B) = (.35) =

14. Read “probability of B given that A occurs”
Ex 5) Suppose I will pick two cards from a standard deck without replacement. What is the probability that I select two spades?
Are the cards independent? NO
P(A & B) = P(A) · P(B|A)
Read “probability of B given that A occurs”
P(Spade & Spade) = 1/4 · 12/51 = 1/17
The probability of getting a spade given that a spade has already been drawn.

15. Ex. 6) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that exactly one bulb is defective?
P(exactly one) = P(D & DC) or P(DC & D)
= (.05)(.95) + (.95)(.05)
= .095

16. Ex. 7) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that at least one bulb is defective?
P(at least one) = P(D & DC) or P(DC & D) or (D & D)
= (.05)(.95) + (.95)(.05) + (.05)(.05)
= .0975

17. Rule 6. At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen.
P(at least 1) = 1 – P(none)

18. What is the probability that at least one bulb is defective?
Ex. 7 revisited) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store.
What is the probability that at least one bulb is defective?
P(at least one) =
1 - P(DC & DC)
.0975

19. Ex 8) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something?
Dr. Pepper
P(at least one winning symbol) =
1 – P(no winning symbols)
= .4686

20. Rule 7: Conditional Probability
A probability that takes into account a given condition

21. Ex 9) In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is If the student is female, what is the probability that she plays sports?

22. Ex 10) The probability that a randomly selected student plays sports if they are male is What is the probability that the student is male and plays sports if the probability that they are male is .49?

23. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
11) What is the probability that the driver is a student?

24. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
12) What is the probability that the driver drives a European car?

25. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
13) What is the probability that the driver drives an American or Asian car?
Disjoint?

26. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
14) What is the probability that the driver is staff or drives an Asian car?
Disjoint?

27. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
15) What is the probability that the driver is staff and drives an Asian car?

28. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
16) If the driver is a student, what is the probability that they drive an American car?
Condition

29. Probabilities from two way tables
Stu Staff Total
American
European
Asian
Total
17) What is the probability that the driver is a student if the driver drives a European car?
Condition

30. What is the probability that an item is returned?
Example 18:
Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items. Construct a tree diagram or a chart for this data.
What is the probability that an item is returned?
If an item is returned, what is the probability that an inexperienced employee assembled it?
Returned
Not returned
Total
Experienced
2.4
77.6 80
Inexperienced
17.6 20
4.8
95.2
100
P(returned) = 4.8/100 = 0.048
P(inexperienced|returned) = 2.4/4.8 = 0.5