1. Agenda Informationer –Uformel evaluering –2 spørgeskemaer + eval. Opsamling fra sidst –Variabel – def. og typer –Fordeling (distribution) –Centrumskøn Sandsynlighedsregning A.Definitioner B.Def. og basale regneregler C.Regneregel og uafhængighed Dagens øvelser

2. Mean (gennemsnittet) The mean is the sum of the observations divided by the number of observations –n betegner antallet af observationer (stikprøvestørrelsen) –y 1, y 2, y 3, … y i,..., y n betegner de n observationer – betegner gennemsnittet It is the center of mass

3. Standard Deviation (standardafvigelsen) Gives a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations. Site Obs. 1 2 3 SumnGns.Std.afv. A55515350,0 B 45615351,0 C 35715352,0

4. A. Learning Objectives 1.Random Phenomena 2.Law of Large Numbers 3.Probability 4.Independent Trials (trail = forsøg / eksperiment) 5.Finding probabilities

5. Learning Objective 1: Random Phenomena For a random phenomena, the outcome is uncertain –In the short-run, the proportion of times that something happens is highly random –In the long-run, the proportion of times that something happens becomes very predictable Probability quantifies long-run randomness

6. Learning Objective 2: Law of Large Numbers As the number of trials increase, the proportion of occurrences of any given outcome approaches a particular number “in the long run” For example, as one tosses a die, in the long run 1/6 of the observations will be a 6. Hvad får vi i det lange løb, hvis vi kaster en terning og bereger andelen, som er større end 3?

7. Learning Objective 3: Probability With random phenomena, the probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations Example: –When rolling a die, the outcome of “6” has probability = 1/6. In other words, the proportion of times that a 6 would occur in a long run of observations is 1/6. Opgave: –Vi tager et kort fra en bunke spillekort bestående af 4 x 13 = 52 kort (og lægger det tilbage igen). Hvor stor en andel af gangene får man et kort over 10 (i det lange løb)?

8. Learning Objective 4: Independent Trials Different trials of a random phenomenon are independent if the outcome of any one trial is not affected by the outcome of any other trial. Example: –If you have 20 flips of a coin in a row that are “heads”, you are not “due” a “tail” - the probability of a tail on your next flip is still 1/2. The trial of flipping a coin is independent of previous flips.

9. Learning Objective 5: How can we find Probabilities? Observe many trials of the random phenomenon and use the sample proportion of the number of times the outcome occurs as its probability. This is merely an estimate of the actual probability. Calculate theoretical probabilities based on assumptions about the random phenomena. For example, it is often reasonable to assume that outcomes are equally likely such as when flipping a coin, or a rolling a die.

10. A. Learning Objectives 1.Random Phenomena 2.Law of Large Numbers 3.Probability 4.Independent Trials (trail=forsøg / eksperiment) 5.Finding probabilities

11. B. Learning Objectives 1.Sample Space (Udfaldsrum) for a Trail (forsøg) 2.Event (Hændelse) 3.Probabilities for a sample space 4.Probability of an event 5.Basic rules for finding probabilities about a pair of events 6.Probability of the union of two events 7.Probability of the intersection of two events

12. Learning Objective 1: Sample Space (udfaldsrum) for a Trail (forsøg) The sample space (udfaldsrummet) is the set of all possible outcomes. Udfaldsrummet for en prøve bestående af 3 spørgsmål, som kan besvares korrekt, C (correct), eller forkert, I, (incorrect) fremgår af figuren. Hvad er udfaldsrummet?

13. Learning Objective 2: Event (hændelse) An event (hændelse) is a subset of the sample space An event corresponds to a particular outcome or a group of possible outcomes. For example; –Event A = student answers all 3 questions correctly = (CCC) –Event B = student passes (at least 2 correct) = (CCI, CIC, ICC, CCC)

14. Learning Objective 3: Probabilities for a sample space Each outcome, f.eks. CCC, in a sample space has a probability The probability of each individual outcome is between 0 and 1. The total (the sum) of all the individual probabilities equals 1.

15. Learning Objective 4: Probability of an Event The Probability of an event A is denoted by P(A) The Probability is obtained by adding the probabilities of the individual outcomes in the event. When all the possible outcomes are equally likely:

16. Learning Objective 4: Eksempel: Forespørgsler på en hjemmeside? Oplist 3 hændelser i ovenstående udfaldsrum. Hvad er ssh. for tilfældigt valgt person... –har kontaktet en hjemmeside med sin mobiltelefon? –har besøgt en hjemmeside med mere end 100.000 besøgende? Hvilken websitestørrelse har størst ssh. for at blive besøgt af en mobiltlf. bruger? Antal siderMobilPCTotal Under 25.0009014.010 25.000-49.9997130.629 50.000-99.9996924.631 100.000 +8010.620 Total

17. Learning Objective 5: Basic rules for finding probabilities about a pair of events Some events are expressed as the outcomes (udfald) that 1.Are not in some other event (complement of the event) 2.Are in one event and in another event (intersection of two events) 3.Are in one event or in another event (union of two events)

18. Learning Objective 5: Complement of an event The complement of an event A consists of all outcomes in the sample space that are not in A. The probabilities of A and of A’ add to 1 P(A ’ ) = 1 – P(A)

19. Learning Objective 5: Disjoint events Two events, A and B, are disjoint if they do not have any common outcomes (udfald)

20. Learning Objective 5: Intersection of two events (fællesmængde) The intersection of A and B consists of outcomes that are in both A and B.

21. Learning Objective 5: Union of two events (foreningsmængde) The union of A and B consists of outcomes that are in A or B or in both A and B.

22. Learning Objective 6: Probability of the Union of Two Events Addition Rule: For the union of two events, P(A or B) = P(A) + P(B) – P(A and B) If the events are disjoint, P(A and B) = 0, so P(A or B) = P(A) + P(B) + 0

23. Learning Objective 6: Example Event A = Mobil Event B = Site med mere end 100.000 sider Hvordan beregner vi P(A and B) til 0,001? Antal siderMobilPCTotal Under 25.0009014.01014.100 25.000-49.9997130.62930.700 50.000-99.9996924.63124.700 100.000 +8010.62010.700 Total31079.89080.200

24. Learning Objective 7: Probability of the Intersection of Two Events Multiplication Rule: For the intersection of two independent events, A and B, P(A and B) = P(A) x P(B) Opgave: Kast med to terninger. Hvad er sandsynligheden for at få to 6’ere? –Definer hændelserne A og B.

25. Learning Objective 7: Example What is the probability of getting 3 questions correct by guessing (= tilfældigheder)? A=correct. Probability of guessing correctly, P(A)=0,2 What is the probability that a student answers at least 2 questions correctly? P( ) + P( ) + P( ) + P( ) = 0,... +... = 0,104

26. Learning Objective 7: Events Often Are Not Independent Don’t assume that events are independent unless you have given this assumption careful thought and it seems plausible. Ø jne ved terningkast M ø ntkast 1-23-45-6 Plat ½ x ⅓ Krone ½ x ⅓

27. Learning Objective 7: Events Often Are Not Independent Example: A Pop Quiz with 2 Multiple Choice Questions –Data giving the proportions for the actual responses –Events: II IC CI CC –Probability: 0,26 0,11 0,05 0,58 –P(CC) = 0,58

28. Learning Objective 7: Events Often Are Not Independent Define the events A and B as follows: –A: {first question is answered correctly} –B: {second question is answered correctly} P(A) = P{(CC), (CI)} = 0.58 + 0.05 = 0.63 P(B) = P{(CC), (IC)} = 0.58 + 0.11 = 0.69 P(A and B) = P{(CC)} = 0.58 If A and B were independent, P(A and B) = P(A) x P(B) = 0.63 x 0.69 = 0.43 Thus, in this case, A and B are not independent!

29. B. Learning Objectives 1.Sample Space (Udfaldsrum) for a Trail (forsøg) 2.Event (Hændelse) 3.Probabilities for a sample space 4.Probability of an event 5.Basic rules for finding probabilities about a pair of events 6.Probability of the union of two events 7.Probability of the intersection of two events

30. C. Learning Objectives 1.Conditional probability 2.Multiplication rule for finding P(A and B) 3.Independent events defined using conditional probability

31. Learning Objective 1: Conditional Probability For events A and B, the conditional probability of event A, given that event B has occurred, is: P(A|B) is read as “the probability of event A, given event B.” The vertical slash represents the word “given”. Of the times that B occurs, P(A|B) is the proportion of times that A also occurs

32. Learning Objective 6: Eksempel: 1) Omregning fra antal til ssh. Antal siderMobilPCTotal Under 25.0009014.01014.100 25.000-49.9997130.62930.700 50.000-99.9996924.63124.700 100.000 +8010.62010.700 Total31079.89080.200 Antal siderMobilPCTotal Under 25.0000,00110,17470,1758 25.000-49.9990,00090,38190,3828 50.000-99.9990,00090,30710,3080 100.000 +0,00100,13240,1334 Total0,00390,99611,0000

33. Learning Objective 1: Example 1 What was the probability of a cell phone visit, given that the site has ≥ 100,000? –Event A: Cell phone is used –Event B: Site has ≥ 100,000

34. Learning Objective 1: Example 1 What is the probability of a cell phone visit given that the site has < 25.000 pages? A = Cell phone is used B = Pages < 25.000 P(A and B) = P(B) = P(A|B) = 0,0063 Antal siderMobilPCTotal Under 25.0000,00110,17470,1758 25.000-49.9990,00090,38190,3828 50.000-99.9990,00090,30710,3080 100.000 +0,00100,13240,1334 Total0,00390,99611,0000

35. Learning Objective 3: Independent Events Defined Using Conditional Probabilities Two events A and B are independent if the probability that one occurs is not affected by whether or not the other event occurs Events A and B are independent if: P(A|B) = P(A), or equivalently, P(B|A) = P(B) If events A and B are independent, P(A and B) = P(A) x P(B)

36. Learning Objective 3: Checking for Independence To determine whether events A and B are independent: –Is P(A|B) = P(A)? –Is P(B|A) = P(B)? –Is P(A and B) = P(A) x P(B)? If any of these is true, the others are also true and the events A and B are independent Ø jne ved terningkast M ø ntkast 1-23-45-6 Plat Krone