1. What is the marginal distribution of income?
What proportion of the sample made over $50,000?

2. Stats: Modeling the World
Chapter 6
From Randomness to Probability

3. Sample Space The set of all possible outcomes Try it!
Albert/Barbara Albert/Carl
Albert/Dianna Barbara/Carl
Barbara/Dianna Carl/Dianna
AND VICE VERSA!!!!
P(Both Girls) = 2/12

4. Basic Probability Rules
For any event A, 0 ≤ P(A) ≤ 1.
The sum of the probabilities for all outcomes in the sample space must equal 1.
P(S) = 1

5. Try it! Is this a possible sample space? Why or why not?
d) {0, 0, 1, 0}
e) {0.1, 0.2, 1.2, -1.5}
Yes No

6. Addition Rule P(A or B) = P(A) + P(B)
provided that A and B are disjoint.
Disjoint (Mutually Exclusive)
The two events, A and B, cannot occur at the same time.

7. Complement Rule
The set of outcomes that are not in the event A is called the complement of A, denoted NOT A.
The probability of an event occurring is
1 minus the probability that it doesn’t
occur: P(A) = 1 – P(NOT A)

8. Try it!

9. Knowledge of event A does not affect the probability of event B
Multiplication Rule
P(A and B) = P(A) x P(B)
provided that A and B are independent.
Independent Events
Knowledge of event A does not affect the probability of event B

10. Try it!

11. Example
The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&Ms, red another 20%, and orange, blue, and green each made up 10%. The rest were brown.
a) If you pick an M&M at random, what is the probability that:
- it is brown?
- it is yellow or orange?
- it is not green?
- it is striped?
30%
20%+10% = 30%
100% - 10% = 90% 0%

12. Example (.3)(.3)(.3) = 0.027 (.8)(.8)(.2) = 0.128 (.8)(.8)(.8) = 0.512
The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&Ms, red another 20%, and orange, blue, and green each made up 10%. The rest were brown.
b) If you pick three M&Ms in a row, what is the probability that:
- they are all brown?
- the third one is the first one that’s red?
- none are yellow?
- at least one is green?
(.3)(.3)(.3) = 0.027
(.8)(.8)(.2) = 0.128
(.8)(.8)(.8) = 0.512
1-No Greens = 1 - (.9)(.9)(.9) = 0.271

13. Example P(0) = 1 – 0.28 = 0.72 P(0 or 1) = 0.72 + 0.17 = 0.89
A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require three or more repairs.
a) What is the probability that a car chosen at random will need:
1. no repairs?
2. no more than one repair?
3. some repairs?
P(0) = 1 – 0.28 = 0.72
P(0 or 1) = = 0.89
P(1 or more) = 1 – P(none) = 0.28

14. Example P(none and none) = (0.72)(0.72) = 0.5184
A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require three or more repairs.
b) If you own two cars, what is the probability that:
1. neither will need repair?
2. both will need repair?
3. at least one will need repair?
P(none and none) = (0.72)(0.72) =
P(some and some) = (0.28)(0.28) =
1 - P(none) = 1 – =

15. Example P(no/no/no) = (0.3)(0.3)(0.3) = 0.027
A certain bowler can bowl a strike 70% of the time. What is the probability that she
a) goes three consecutive frames without a strike?
b) makes her first strike in the third frame?
c) has at least one strike in the first three frames
d) bowls a perfect game (12 consecutive strikes)
P(no/no/no) = (0.3)(0.3)(0.3) = 0.027
P(no/no/yes) = (0.3)(0.3)(0.7) = 0.063
1 - P(no/no/no) = 1 – = 0.973
P(12 strikes) = (0.7)^12 =

16. Example P(RRR) = (0.29)(0.29)(0.29) = 0.024
Suppose that in your city 37% of voters are registered Democrats, 29% Republicans, and 11% other parties. Voters not aligned with any party are termed “independent”. You are conducting a poll by calling registered voters at random. In your first three calls, what is the probability you talk to:
a. All Republicans?
b. No Democrats?
c. At least one Independent?
P(RRR) = (0.29)(0.29)(0.29) = 0.024
P(ND/ND/ND) = (0.63)(0.63)(0.63) = 0.250
1 – P(no Indy)= 1 - (0.89)(0.89)(0.89) = 0.295

17. P(40 or 100) = 3/12 = 0.25
P(10/10) = (0.5)(0.5) = 0.25
P(20/20/20) = (0.25)(0.25)(0.25) =

18. P(20 or less 4 times) = (0.75)(0.75)(0.75)(0.75) = 0.316
P(No/No/No/No/WIN) = (11/12)^4 * (1/12) =
1 - P(no gold in 6) = 1 – (11/12)^6 = 0.407

19. Disjoint events CANNOT be independent!!
Common Errors
Don’t confuse disjoint and independent!!
Disjoint events CANNOT be independent!!

20. Example
In a previous WAP problem, we calculated the probabilities of getting various M&Ms.
a) If you draw one M&M, are the events of getting a red one and getting an orange one disjoint or independent or neither?
b) If you draw two M&Ms one after the other, are the events of getting a red on the first and a red on the second disjoint or independent or neither?
c) Can disjoint events ever be independent? Explain.