Probability & Statistical Inference Lecture 2

1. Probability & Statistical Inference Lecture 2 MSc in Computing (Data Analytics)

2. Lecture Outline • Introduction • Introduction to Probability Theory • Discrete Probability Distributions • Question Time

3. Introduction

4. Probability & Statistics • We want to make decisions based on evidence from a sample i.e. extrapolate from sample evidence to a general population • To make such decisions we need to be able to quantify our (un)certainty about how good or bad our sample information is. MakeInference Describe

5. Probability & Statistics - Example • Example: How many voters will give F.F. a first preference in the next general election ? • researcher A takes a sample of size 10 and find 4 people who say they will • researcher B takes a sample of size 100 and find 25 people who say they will • Researcher A => 40% • Researcher B => 25% • Who would you believe?

6. Probability & Statistics - Example • Intuitively the bigger sample would get more credence but how much better is it, and are either of the samples any good? • Probability helps • Descriptive Statistics are helpful but still lead to decision making by 'intuition‘ • Probability helps to quantify (un)certainty which is a more powerful aid to the decision maker

7. Probability & Statistics • Using probability theory we can measure the amount of uncertainty/certainty in our statistics.

8. Intuitions and Probability – Lotto example • If you had an Irish lotto ticket which of these sets of numbers is more likely to win: • 1 2 3 4 5 6 Odds of winning are 1 in 8145060 • 2 11 26 27 35 42 Odds of winning are 1 in 8145060

9. Intuitions and Probability – Disease example • Suppose we have a diagnostic test for a disease which is 99% accurate. • A person is picked at random and tested for the disease • The test gives a positive result. What is the probability that the person actually has the disease? • 99% ?

10. Disease example No!! IT depends on how common or rare the disease is. Suppose the disease affects 1 person in 10,000 Of those who test positive onlyhave the disease

11. Introduction to Probability Theory

12. Some Definitions • An experimentthat can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. • The set of all possible outcomes of a random experiment is called the sample space of an experiment and is denote by S • Example: • Experiment: Toss two coins and observe the up face on each • Sample Space: • Observe HH • Observe HT • Observe TH • Observe TT S : {HH,HT,TH,TT}

13. Some Definitions • A sample space is discrete if it consists of a finite or countable infinite set if outcomes. • A sample space is continuous if it contains an interval or real numbers. • An event is a subset of the sample space of a random experiment.

14. Counting • A permutation of the elements is an ordered sequence of the elements. • Example: S : {a,b,c} • All the permutations of the elements of S are abc, acb, bca, bac, cba & cab. • The number of permutations of n different elements is n! Where: n! = n * (n-1) * (n-2) * .......* 2 * 1 Above n=3 => 3! = 3 * 2 * 1 = 6

15. Counting • The number of permutations of subsets r elements selected from a set of n different elements is • Where order is not important when selecting r elements from a set of n different elements is called a combination:

16. Probability • Whenever a sample space consists of N Possible outcomes that are equally likely, the probability of the outcome 1/N. • For a discrete sample space, the probability of an event E, denoted by P(E), equals the sum of the probabilities of the outcome in E. • Some rules for probabilities: • For a given sample spce containing n event sE1, E2, E3, ........,En • All simple event probabilities must lie between 0 and 1: 0 <= P(Ei) <= 1 for i=1,2,........,n • The sum of the probabilities of all the simple events within a sample space must be equal to 1:

17. Probability – Example 1 • Example: • Experiment: Toss two coins and observe the up face on each • Sample Space: S : {HH,HT,TH,TT} • Probability of each event: • E = HH => P(HH) = 1/4 • E = HT => P(HT) = 1/4 • E = TH => P(TH) = 1/4 • E = TT => P(HH) = 1/4

18. Probability – Example 1 • The probability of an event A is equal to the sum of all the probabilities in event A: • Example: • Experiment: Toss two coins and observe the up face on each • Event A: {Observe exactly one head} P(A) = P(HT) + P(TH) = ¼ + ¼ = ½ • Event B : {Observe at least one head} P(B) = P(HH) + P(HT) + P(TH) = ¼ + ¼ + ¼ = ¾

19. Probability – Example 2 • 5 card poker example. It doesn't matter in poker in what order the cards are dealt, merely the combination of certain cards wins. So, how many different hands of poker are there? = ? • What is the probability of being dealt a royal straight flush? • What is the probability of being dealt a poker (i.e. 4 of a kind)?

20. Compound Events • The union of two event A and B is the event that occurs if either A or B, or both, occur on a single performance of the experiment denoted by A U B (A or B) • The intersection of two events A and B is the event that occurs if both A and B occur on a single performance of an experiment denoted by A B or (A and B) • Example: Consider a die tossing experiment with equally likely simple events {1,2,3,4,5,6}. Define the events A, B and C. • A:{Toss an even number} = {2,4,6} • B:{Toss a less than or equal to 3} = {1,2,3} • C:{Toss a number greater than 1} = {2,3,4,5,6} • Find:

21. Complementary Event • The complementary of an event A is the event that A does not occur denoted by A´ • Note that AU A` = S, the sample space • P(A) + P(A`) =1 => P(A) = 1 – P(A`)

22. Questions • What is the sample space when a coin is tossed 3 times? • What is the probability of tossing all heads or all tails. • What is the sample space of throwing a fair die. • If a fair die is thrown what is the probability of throwing a prime number (2,3,5)?

23. Questions • A factory has two assembly lines, each of which is shut down (S), at partial capacity (P), or at full capacity (F). The following table gives the sample space For where (S,P) denotes that the first assembly line is shut down and the second one is operating at partial capacity. What is the probability that: • Both assembly lines are shut down? • Neither assembly lines are shut down • At least one assembly line is on full capacity • Exactly one assembly line is at full capacity

24. Conditional Probability • The conditional probability of event A conditional on event B is for P(B)>0. It measures the probability that event A occurs when it is known that event B occur. • Example: A = odd result on die = {1,3,5} • B = result > 3 = {4,5,6}

25. Conditional Probability Example • Example: A study was carried out to investigate the link between people’s lifestyles and cancer. One of the areas looked at was the link between lung cancer and smoking. 10,000 people over the age of 55 were studied over a 10 year period. In that time 277 developed lung cancer. What is the likelihood of somebody developing lung cancer given that they smoke?

26. Conditional Probability Example • Event A: A person develops lung cancer Event B: A person is a smoker P(A) = 277/10,000 = 0.027 P(B) = 3,566/10,000 = 0.356

27. Exercises • A ball is chosen at random from a bag containing 150 balls that are either red or blue and either dull or shinny. There are 36 red, shiny balls and 54 blue balls. There are 72 dull balls. • What is the probability of a chosen ball being shiny conditional on it being red? • What is the probability of a chosen ball being dull conditional on it being blue?

28. Mutually Exclusive Events • Two events, A and B, are mutually exclusive given that if A happens then B can’t also happen. • Example: Roll of a die A = less than 2 B = even result There is no way that A and B can happen at the same time therefore they are mutually exclusive events

29. Rules for Unions • Additive Rule: • Additive Rule for Mutually Exclusive Events

30. Example • Records at an industrial plant show that 12% of all injured workers are admitted to hospital for treatment, 16% are back on the job the next day, and 2% are both admitted to a hospital for treatment and back to work the next day. If a worker is injured what is the probability that the worker will be either admitted to hospital or back on the job the next day or both?

31. Independent Events • Events A and B are independent if it is the case that A happening does not alter the probability that B happens. • Example : A = even result on die B = result > 2 • Then, let us say we are told the result on the die (which someone has observed but not us) is even so knowing this, what is the probability that the event B has happened? Sample space: {2, 4, 6} B = 4 or 6 => P(B) = 2/3

32. Independent Events • But if we didn’t know about the even result we would get: Sample space: {1, 2, 3, 4, 5, 6} B = 3 or 4 or 5 or 6 => P(B) = 4/6 = 2/3 so knowledge about event A has in no way changed out probability assessment concerning event B

33. Rules for Intersection • Multiplicative Rule of Probability • If events A and B are independent then

34. Bayes Theorem • One of a number of very useful results: - here is simplest definition: • Suppose: You have two events which are ME and exhaustive – i.e. account for all the sample space – • Call these events A and event (read ‘not A’). • Further suppose there is another event B, such that • P(B|A) > 0 and P(A|B) > 0. • Then Bayes theorem states:

35. Discrete Probability Distributions

36. Discrete Random Variable • A Random Variable (RV) is obtained by assigning a numerical value to each outcome of a particular experiment. • Probability Distribution: A table or formula that specifies the probability of each possible value for the Discrete Random Variable (DRV) • DRV: a RV that takes a whole number value only

37. Example: What is the probability distribution for the experiment to assess the no of tails from tossing 2 coins; • Sample Space • Coin 1Coin 2 • T T • T H • H T • H H • x = no. of tails is the RV • xP(x) • 0 = P(HH) = 0.25 • 1 = P(TH) + P(HT) = 0.50 • 2 = P(TT) = 0.25 • P( any other value ) = 0 • N.B.  P(x) = 1 • 0  P(x)  1 for all values of x

38. Mean of a Discrete Random Variable • Mean of a DRV =  = Σx * p(x) • Example: Throw a fair die • xP(x) x * P(x) • 1 0.1667 0.17 • 2 0.1667 0.33 • 3 0.1667 0.50 • 4 0.1667 0.67 • 5 0.1667 0.83 • 6 0.1667 1.00 • P(any other value) = 0 0 • Mean =  =Σx * p(x) = 3.5

39. Some Examples Simulated Sample size = 10 mean = 3.7 S.D. = 2.1 Simulated Sample size = 100 mean = 3.54 S.D. = 1.67

40. Simulated Sample size = 1000 Mean = 3.49 S.D. = 1.73

41. Note: • The largest simulation had the mean closest to that predicted by the probability distribution • As the simulations got bigger the mean approached 3.5 • Mean of DRV is the mean of a large number of independent experiments (trials).

42. Standard Deviation of a DRV

43. Example: Rolling one die xP(x) x2 * P(x) 1 0.1667 0.17 2 0.1667 0.67 3 0.1667 1.50 4 0.1667 2.67 5 0.1667 4.17 6 0.1667 6.00 P(any other value) = 0 0 = 15.17 15.17 - (3.5)2 = 15.17 - 12.25 = 2.92 => S.D. = 1.71 • Simulations: • N=10 => S.D. = 2.1 • N=100 => S.D. = 1.67 • N=1,000 => S.D. = 1.73

44. Binomial (Probability) Distribution • Many experiments lead to dichotomous responses (i.e. either success/failure, yes/no etc.) • Often a number of independent trials make up the experiment • Example: number of people in a survey who agree with a particular statement? Survey 100 people => 100 independent trials of Yes/No The random variable of interest is the no. of successes (however defined) • These are Binomial Random Variables

45. Binomial Distribution Example 4 people tested for the presence of a particular gene. success = presence of gene P(gene present / success) = 0.55 P(gene absent / failure) = 0.45 P(3 randomly tested people from 4 have gene)? Assume trials are independent - e.g. the people are not related There is 4 ways of getting 3 successes

46. Binomial Distribution Example • Using Independence rule we can calculate the probability of each outcome: • Outcome 1: 0.55  0.55  0.55  0.45 = 0.07486875 Outcome 2: 0.55  0.55  0.45  0.55 = 0.07486875 Outcome 3: 0.55  0.45  0.55  0.55 = 0.07486875 Outcome 4: 0.45  0.55  0.55  0.55 = 0.07486875 4 ways of getting result each with P=0.07486875 => 4  0.07486875 = 0.299475 => P(3 randomly tested people have gene) = 0.299475

47. Binomial Distribution Example • A more convenient way of mathematically writing the same result is as follows: • the number of ways you can get three successes from 4 trials is a combination:

48. Binomial Distribution – General Formula • This all leads to a very general rule for calculating binomial probabilities: In General Binomial (n,p) n = no. of trials p = probability of a success x = RV (no. of successes) • Where P(X=x) is read as the probability of seeing x successes.

50. Binomial Distribution • For all binomials the mean is given by the simple formula;  = n  p • Example: from previous example  = 4  0.55 = 2.2 • Standard deviation also has simple formula for all Binomials • Example: from previous example = 0.995