advertisement

5/17/2010 Statistics 111 - Lecture 1 Intro to Probability “The probable is what usually happens.” (Aristotle ) Moore, McCabe and Craig: Section 4.1,4.2,4.3,4.5 Deterministic vs. Random Processes • In deterministic processes, the outcome can be predicted exactly in advance • In random processes, the outcome is not known exactly, but we can still describe the probability distribution of possible outcomes • Imagine tossing a coin -- if we knew exactly the angle of the throw, the initial force and the air friction we’d know exactly if the coin would lend on Heads or Tails. • However we don’t know all of these things but we usually say that there is 50% chance to get either. 1 5/17/2010 Definition: Event • An event is an outcome or a set of outcomes of a random process Example: Tossing a coin three times Event A = getting exactly two heads = {HTH, HHT, THH} Example: Picking real number X between 1 and 20 Event A = chosen number is at most 8.23 = {X ≤ 8.23} Example: Tossing a fair dice Event A = result is an even number = {2, 4, 6} • • Notation: P(A) = Probability of event A Probability Rule 1: 0 ≤ P(A) ≤ 1 for any event A Sample Space • The sample space S of a random process is the set of all possible outcomes Example: one coin toss S = {H,T} Example: three coin tosses S = {HHH, HTH, HHT, TTT, HTT, THT, TTH, THH} Example: roll a six-sided dice S = {1, 2, 3, 4, 5, 6} • Probability Rule 2: The probability of the whole sample space is 1 P(S) = 1 2 5/17/2010 Combinations of Events • The complement Ac of an event A is the event that A does not occur • Probability Rule 3: P(Ac) = 1 - P(A) • The union of two events A and B is the event that either A or B or both occurs • The intersection of two events A and B is the event that both A and B occur Event A Complement of A Union of A and B Intersection of A and B Disjoint Events • Two events are called disjoint if they have no outcomes in common and can never happen together • Example: coin is tossed twice • S = {HH,TH,HT,TT} • Events A={HH} and B={TT} are disjoint • Events A={HH,HT} and B = {HH} are not disjoint • Probability Rule 4: If A and B are disjoint events then P(A or B) = P(A) + P(B) 3 5/17/2010 Independent events • Events A and B are independent if the probability that A occurs is not affected or changed by the occurrence of the B. • Example: tossing two coins Event A = first coin is a head Event B = second coin is a head Independent • Probability Rule 5: If A and B are independent P(A and B) = P(A) x P(B) Equally Likely Outcomes Rule • If all possible outcomes from a random process have the same probability, then 𝑃(𝐴) = # of outcomes in A # of outcomes in S • Example: One Dice Tossed 𝑃 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 = "2" + "4" + "6" 1 = "1" + "2" + "3" + "4" + "5" + "6" 2 • Note: equal outcomes rule only works if the number of outcomes is “countable” • Eg. of an uncountable process is sampling any fraction between 0 and 1. Impossible to count all possible fractions ! 4 5/17/2010 Equally Likely Outcomes Rule • You toss two dice. What is the probability of the outcomes summing to 5? • 1. Find the sample space. • 2. Define the event of interest (A) • 3. Count the number of outcomes in A Combining Probability Rules Together • Initial screening for HIV in the blood first uses an enzyme immunoassay test (EIA) • Even if an individual is HIV-negative, EIA has probability of 0.006 of giving a positive result • Suppose 100 people are tested who are all HIV-negative. What is probability that at least one will show positive on the test? • First, use complement rule: P(at least one positive) = 1 - P(all negative) 5 5/17/2010 Combining Probability Rules Together • Now, we assume that each individual is independent and use the multiplication rule for independent events: P(all negative) = P(test 1 negative) ×…× P(test 100 negative) • P(test negative) = 1 - P(test positive) = 0.994 P(all negative) = 0.994 ×…× 0.994 = (0.994)100 • So, we finally we have P(at least one positive) =1− (0.994)100 = 0.452 Conditional Probabilities • Eg. imperfect diagnostic test for a disease Disease + Disease - Total Test + 30 10 40 Test - 10 50 60 Total 40 60 100 • What is probability that a person has the disease? Answer: 40/100 = 0.4 • What is the probability that a person has the disease given that they tested positive? More Complicated ! 6 5/17/2010 Definition: Conditional Probability • Let A and B be two events in sample space • The conditional probability that event B occurs given that event A has occurred is: 𝑃 𝐵𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴) • Eg. probability of disease given test positive 𝑃 𝑑𝑖𝑠𝑒𝑎𝑠𝑒 + 𝑡𝑒𝑠𝑡 + = 𝑃(𝑑𝑖𝑠𝑒𝑎𝑠𝑒 𝑎𝑛𝑑 𝑡𝑒𝑠𝑡+) 30 100 = = 0.75 𝑃(𝑡𝑒𝑠𝑡+) 40 100 Independent vs. Non-independent Events • If A and B are independent, then P(A and B) = P(A) x P(B) which means that conditional probability is: 𝑃 𝐵𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴) × 𝑃(𝐵) = = 𝑃(𝐵) 𝑃(𝐴) 𝑃(𝐴) • We have a more general multiplication rule for events that are not independent: P(A and B) = P(B | A) × P(A) 7 5/17/2010 Conditional Probabilities • The (famous) Three Door Problem: • Let’s say you picked door # 1: The host can open either door 2/3 The host must open door number 3 The host must open door number 2 If you’ll switch you’ll loose If you’ll switch you’ll win If you’ll switch you’ll win The host gave you extra information by opening the door Break June 3, 2008 Stat 111 - Lecture 6 - Probability 16 8