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 people23304060 Probability.507.706.891.994

3. Probability Formal study of uncertainty The engine that drives Statistics Primary objective of lecture unit 4: 1.use the rules of probability to calculate appropriate measures of uncertainty. 2.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. A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Randomness and probability Randomness ≠ chaos

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). First series of tosses Second series The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

7. 4.1 The Laws of Probability 1.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 Approaches to Probability

8. Approaches to Probability (cont.) 2.Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc. 3.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  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} 3.Select 1 card from a deck of 52 cards. S = {all 52 cards}

12. Laws of Probability

13. Coin Toss Example: S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5 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 Probability rules (cont’d) 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 people23304060 Probability.507.706.891.994

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 A  A 

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

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

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 ColorBrownRedYellowGreenOrangeBlue Probability0.30.2 0.1 ? 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? What is the probability that a random M&M is any of red, yellow, or orange? 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 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1 P(red or yellow or orange) = P(red) + P(yellow) + P(orange) = 0.2 + 0.2 + 0.1 = 0.5

23. Example: toss a fair die once S = {1, 2, 3, 4, 5, 6} 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 + 3/6 - 1/6 = 5/6

24. End of First Part of Section 4.1