1. Unit 4 Probability Basics
Laws of Probability
Odds and Probability
Probability Trees

2. Birthday Problem
What is the smallest number of people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
Answer: 23
No. of people
Probability

3. Probability Formal study of uncertainty
The engine that drives Statistics
Primary objective of lecture unit 4:
use the rules of probability to calculate appropriate measures of uncertainty.
Learn the probability basics so that we can do Statistical Inference

4. Introduction Nothing in life is certain
We gauge the chances of successful outcomes in business, medicine, weather, and other everyday situations such as the lottery or the birthday problem
Tomorrow's Weather

5. Randomness and probability
Randomness ≠ chaos
A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

6. Coin toss
The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip).
The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
First series of tosses
Second series

7. 4.1 The Laws of Probability
Approaches to Probability
Relative frequency
event probability = x/n,
where x=# of occurrences of event of interest, n=total # of observations
Coin, die tossing; nuclear power plants?
Limitations
repeated observations not practical

8. Approaches to Probability (cont.)
Subjective probability
individual assigns prob. based on personal experience, anecdotal evidence, etc.
Classical approach
every possible outcome has equal probability (more later)

9. Basic Definitions
Experiment: act or process that leads to a single outcome that cannot be predicted with certainty
Examples:
1. Toss a coin
2. Draw 1 card from a standard deck of cards
3. Arrival time of flight from Atlanta to RDU

10. Basic Definitions (cont.)
Sample space: all possible outcomes of an experiment. Denoted by S
Event: any subset of the sample space S;
typically denoted A, B, C, etc.
Null event: the empty set F
Certain event: S

11. Examples 1. Toss a coin once S = {H, T}; A = {H}, B = {T}
2. Toss a die once; count dots on upper face
S = {1, 2, 3, 4, 5, 6}
A=even # of dots on upper face={2, 4, 6}
B=3 or fewer dots on upper face={1, 2, 3}
Select 1 card from a
deck of 52 cards.
S = {all 52 cards}

12. Laws of Probability

13. Probability rules (cont’d)
Coin Toss Example:
S = {Head, Tail}
Probability of heads = 0.5
Probability of tails = 0.5
Probability rules (cont’d)
3) The complement of any event A is the event that A does not occur, written as A.
The complement rule states that the probability of an event not occurring is 1 minus the probability that is does occur.
P(not A) = P(A) = 1 − P(A)
Tail  = not Tail = Head
P(Tail ) = 1 − P(Tail) = 0.5 
Venn diagram:
Sample space made up of an event A and its complement A , i.e., everything that is not A.

14. Birthday Problem
What is the smallest number of people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
Answer: 23
No. of people
Probability

15. Example: Birthday Problem
A={at least 2 people in the group have a common birthday}
A’ = {no one has common birthday}

16. Unions: , or Intersections: , and
AÇB
AÈB

17. Mutually Exclusive (Disjoint) Events
Venn Diagrams
Mutually Exclusive (Disjoint) Events
A and B disjoint: A  B=
Mutually exclusive or
disjoint events-no outcomes
from S in common
AÇB
AÈB
A and B not disjoint

18. Addition Rule for Disjoint Events
4. If A and B are disjoint events, then P(A or B) = P(A) + P(B)

19. Laws of Probability (cont.)
General Addition Rule
5. For any two events A and B
P(A or B) = P(A) + P(B) – P(A and B)

20. General Addition Rule _ For any two events A and B +
P(A or B) = P(A) + P(B) - P(A and B)
P(A) =6/13
A or B + A
P(B) =5/13 _ B
P(A and B) =3/13
P(A or B) = 8/13

21. Laws of Probability: Summary
1. 0  P(A)  1 for any event A
2. P() = 0, P(S) = 1
3. P(A’) = 1 – P(A)
4. If A and B are disjoint events, then
P(A or B) = P(A) + P(B)
5. For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)

22. M&M candies
If you draw an M&M candy at random from a bag, the candy will have one of six colors. The probability of drawing each color depends on the proportions manufactured, as described here:
What is the probability that an M&M chosen at random is blue?
Color
Brown
Red
Yellow
Green
Orange
Blue
Probability
0.3
0.2
0.1 ?
S = {brown, red, yellow, green, orange, blue}
P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1
P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)]
= 1 – [ ] = 0.1
What is the probability that a random M&M is any of red, yellow, or orange?
P(red or yellow or orange) = P(red) + P(yellow) + P(orange)
= = 0.5

23. Example: toss a fair die once
A = even # appears = {2, 4, 6}
B = 3 or fewer = {1, 2, 3}
P(A or B) = P(A) + P(B) - P(A and B)
=P({2, 4, 6}) + P({1, 2, 3}) - P({2})
= 3/ / /6 = 5/6

24. End of First Part of Section 4.1