Lecture 2: Probability. Readings: Sections Statistical Inference: drawing conclusions about the population based on a sample

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1 Lecture 2: Probability Readings: Sections Introduction Statistical Inference: drawing conclusions about the population based on a sample Parameter: a number that describes the population a fixed number but in practice we do not know its value Statistic: a number that describes a sample its value is known when we have taken a sample, but it can change from sample to sample (sampling variability) Sample statistics serve as estimates of the population parameters. For example, sample mean x vs. population mean µ sample standard deviation s vs. population standard deviation σ Example 1: A survey by a market research firm asked a nationwide random sample of 2500 adults if they agreed or disagreed that I like buying new clothes, but shopping is often frustrating and time-consuming. Of the respondents, 1650, or 66%, said they agreed. The sample proportion who agree is ˆp = = 66% Statistic: ˆp Parameter: proportion (call it p) of all U.S. adults who would have said Agree if asked the same question Question: How trustworthy is an inference procedure? What would happen is we repeated it many times? 1

number of repetitions Probability of an outcome: proportion of times the outcome would occur in a very long series of repetitions

2 2 Probability 2.1 Randomness Example 2: Flip a coin. What is the chance of getting heads? Randomness: a phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular pattern of outcomes in a large number of repetitions Probability of an outcome: proportion of times the outcome would occur in a very long series of repetitions Probability model contains A list of all possible outcomes A probability for each outcome 2.2 Sample space and events Sample space (S): the set of all possible outcomes Event: a subset of the sample space 2

Example 3: Identify the sample space for the following Toss a coin once: S = {H, T } Toss a coin three times: S = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T } Select one card from a deck of 52

3 Example 3: Identify the sample space for the following Toss a coin once: S = {H, T } Toss a coin three times: S = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T } Select one card from a deck of 52 cards and record the suit: S = {diamond, heart, clud, spade} Example 4: Take a deck of cards and discard all but the face cards. Shuffle the cards thoroughly and draw the top card. Sample space: S = {JS, JC, JH, JD, QS, QC, QH, QD, KS, KC, KH, KD} Let A represent the event that you draw a king A = {KS, KC, KH, KD} Let B represent the event that you draw a heart A = {JH, QH, KH} 2.3 Set operations Intersection : event A B (A AND B) contains the common points of A and B Example 4 (cont d): A B = {KH} n A i = A 1 A2... An i=1 Union : event A B (A OR B) contains all points in A and B 3

4 Example 4 (cont d): A B = {KS, KC, KH, KD, JH, QH} n A i = A 1 A2... An i=1 Complement : event A (NOT A) contains all points that are not in A (event A occurs when A does not occur) Example 4 (cont d): A = {JS, JC, JH, JD, QS, QC, QH, QD} A c is another commonly used notation for the complement of A Mutually exclusive or disjoint events: two events A and B are mutually exclusive or disjoint if they have no common outcomes Example 4 (cont d): Let E be the event of drawing a red Jack and F be the event of drawing a King of spade E = {JH, JD} F = {KS} Are E and F mutually exclusive? 2.4 Probability rules 1. The probability of an event A, denoted as P (A), satisfies 0 P (A) 1 2. The probability of the sample space S, P (S), is equal to 1 3. Addition rule for disjoint events: If events A and B are mutually exclusive, then P (A B) = P (A) + P (B) 4

5 More generally, for a sequence of mutually exclusive events A 1, A 2,..., A n, n P ( A i ) = P (A 1 ) + P (A 2 ) P (A n ) i=1 4. General addition rule: For any events A and B, which need not be mutually exclusive, P (A B) = P (A) + P (B) P (A B) 5. Complement rule: P (A ) = 1 P (A) 6. Empty set has zero probability, i.e., P ( ) = 0 7. If events A and B are mutually exclusive, then P (A B) = 0 Example 5: In Lafayette, 32% of households subscribe to the Indianapolis Star, 49% of household subscribe to the Lafayette Journal and Courier, and 12% of households subscribe to both papers. Select one household at random, let A be the event that the selected household subscribes to the Indianapolis Star and B be the event that the selected household subscribes to the Lafayette Journal and Courier. What is the probability that this household does not subscribe either paper? P ((A B) ) = 1 P (A B) = 0.31 What is the probability that this household subscribes only to the Lafayette Journal and Courier? P (A B ) = 0.2 What is the probability that this household subscribes to only one paper? P ((A B ) (A B)) = =

6 2.5 Equally likely outcomes Example 6: Roll a balanced, six-sided die. What is the probability that you roll a number less than 3? Equally likely outcomes: If an experiment has N possible outcomes, all equally likely, then each individual outcome has probability 1/N. The probability of any event A is P (A) = Number of outcomes in A Number of outcomes in S = N(A) N Example 7: Toss a fair coin three times. What is the probability that there is exactly one head? Sample space S = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T } E = exactly one head = {HT T, T HT, T T H} P (E) = 3 8 = Conditional Probability Example 8: Draw a card from a deck of 52 cards. What is the probability that the card is a heart? 1/4 Suppose that we know the card is red, what is the probability that it is a heart? 1/2 Suppose that we know the card is black, what is the probability that it is a heart? 0 Conditional probability of A given B: P (A B) = P (A B) P (B) Conditional probability is a way to incorporate additional information Conditional probability is undefined when P (B) = 0 Example 9: I have an unbalanced die with probabilities: prob Roll the die once and record the number. Let A = the event that the number rolled is even, B = the event that the number rolled is less than 4. 6

7 P (A B) P (A B) = = 0.02 P (B) 0.13 = P (B A) = P (B A) P (A) = = P (B A ) = P (B A ) P (A ) = = Example 10: The prevalence of HIV in a population is 2%. A certain test for HIV has sensitivity 99.7% and specificity 98.5%. This means that if a person is HIV+, the probability that the test is positive is And if a person is HIV, the probability that the test is negative is What is the probability that a person chosen at random from the population will test positive? P (test+) = P (HIV+ and Test+) + P (HIV- and Test+) = = If a person tests positive, what is the probability that he/she is really HIV positive? P (HIV+ Test+) = P (HIV+ and Test+) P (Test+) = = Independence Two events A and B are said to be independent if a. P (A B) = P (A) b. P (B A) = P (B) c. P (A B) = P (A)P (B) If one of these three is true, all three are true Example 11: Flip a fair coin twice. Let A = getting heads on the first toss and B = getting heads on the second toss. Are A and B independent? 7

8 Example 12: Toss a balanced die three times. What is the probability that all three numbers are greater than four? Independence vs. mutually exclusiveness: If two events are mutually exclusive, they are not independent. 8

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)