Probability

1. Probability • The tool that allows the statisticians to use sample information to make inferences about or to describe the population from which the sample was drawn

2. Some concepts *Experiment: is the process by which an observation is obtained. Examples: - toss a die is a experiment - Draw a card from a deck of card *Event: is the outcome of an experiment Examples: - toss a die: observe a 1 observe an odd number (1,3,5)

3. *Simple Event: an event that cannot be decomposed is called a simple event Examples: -toss a die, event that observe a 1 is a simple event. event that observe an odd number is not - How many simple events are there in the card experiment? ** Each experiment will result in one and only one of the simple events. ** An event is a collection of one or more simple events - the event of observing an odd number is made up of 3 simple events

4. * Sample Space: is the set of all possible direct outcomes of an experiment Example: - tossing a dice, sample space (1,2,3,4,5,6) - tossing two dice?

5. Formal Definition of Probability: If you repeat an experiment over and over again say (N) times and event A is observed n times, then we view the probability of event as: • P(A)= n / N • **practically, we cannot get the probability by using this method. One of approaches we are going to learn to calculate the probability is called Simple Events Approach • # of simple events included in A • P(A)= • # of simple events included in sample space. • # of possible direct outcomes in A • P(A)= • # of all possible direct outcomes of the experiment

6. Property of simple event probability • Each probability must lie between 0 and 1, inclusive. (any number negative or greater than 1 is not probability) • The sum of the probabilities for all simple events in S equals to 1 (the probability of sample space is 1)

7. Example Experiment - toss a coin twice (H: head T: tail) Step one – list all simple event, table or tree Step two – identify the sample space Step three – calculate probabilities What is the probability that we observe (H,H) observe the same type observe the different type observe at least one head

8. Problem: toss the coin three times. *draw tree diagram *how many simple events can we have? *what is sample space? *what is prob of following events: observe (H H T) observe same type three times. observe at most one head

9. Learning to Count (all over again) • Playing the lottery, how many different possible combinations are there for the winning number • How many different ways can a 8 person batting order be arranged with 9 players • How many different combinations of cards could be drawn in Blackjack

10. How to count when you are picking r things out of n • Two questions you have to ask • Can there be repeats or replacements • Does the order matter • Results • There can be repeats and order matters: n^r • There can not be repeats and order matters: permutations • There can not be repeats and order does not matter: combination • There can be repeats and order does matter: not talked about yet

11. Combination and Permutation Pick r things out of n at a time, can not have repeats Permutation formula: If position (order) does matter, we have following number of choices: Combination formula: If position (order) does not matter, we have following number of choices:

12. Intuition of combination and permutation formula A permutation can be thought of a selection process in which objects are selected one by one in a certain order. If we want to select r objects out of n, the number of first possible choice is n, and then next is n-1 (because they cannot be the same objects more than once), and then n-2, finally (n-r+1). so the choices are n*(n-1)*(n-2)…*(n-r+1) Given one combination(select r out of n, no matter what order they are), we should have r! number of possible permutations. So the number of combination is number of permutation divided by r!

13. Problem: codes edition • Choose 4 numbers from 0—9 to comprise a 4-digit code. • If number cannot be the same, how many codes can we have? • If number can be the same, how many codes can we have?

14. Problem: Cold Stone Ice Cream • Suppose Cold Stone provide 7 different types of toppings to put in your ice cream • If you have enough money to choose three different toppings and the order those toppings go in does not matter, how many choices do you have? • If you have enough money to choose 4 different toppings and the order those toppings go in does not matter, how many choices do you have?

15. Problem: Coach’s choice • Ol’ Roy must choose 5 out of 10 players to put on his starting lineup. • If players’ position does not matter, how many choices the coach has? • If position does matter, how many choices he has?

16. Event composition and Event relations * Intersection of event A and B, denoted by AB or ( ), is the event that both A and B occur Example: toss a die, what is AB for following: 1) A={1}, B={observe odd number} 2) A={observe odd number}, B={even number} 3) A={number greater than 4}, B=Sample space

17. Event Composition and Relations * Union of event A and B, denoted by A+B ( ), is the event that A or B or both occur. Example: toss a die, what is A+B 1) A={1}, B={observe odd number} 2) A={observe odd number}, B={even number} 3) A={number greater than 4}, B=Sample space

18. Event Composition and Relations * Complement of an event A, denoted by consists of the all the simple event in the sample space that are not in A. Example: given following A, what is A complement 1) A={1} 2) A={observe odd number} 3) A=sample space

19. Event Composition and Relations * Mutually exclusive/ Disjoint events: two events A and B are said to be mutually exclusive if when A occurs, B cannot occur(and visa versa). • This implies that A and B have no common parts and = empty set( ) )

20. Property of disjoint events 1) P(AB) = 0 2) P( ) = P(A) + P(B) S A B

21. Example: toss a die • A={observe odd number}. • B={observe even number} • Are they disjoint events? • Use simple events approach to calculate P(A), P(B), P( ) and P( )

22. Additive Rule of Probability • - Given two events A and B, • P( A+B ) = P( A ) + P( B ) - P( AB ) • If A and B are mutually exclusive, then P(AB)=0 and • P(A+B) = P ( A )+ P ( B )

23. Example: toss the coin twice A={observe at least one head}. B={observe at least one tail} Define events A, B, AB, and as collection of simple events. Use simple events approach to calculate the probability of these events. Use additive rule of probability to calculate the P( )

24. Another important of relationship * Given any event A, P(A) + P ( ) = 1

25. Example: toss a die twice What is probability of sum of two observation greater than 3.

26. Conditional Probability * Conditional probability of event B given A has occurred is Example: toss a die what is probability of observing a 1 given we already observed an odd number on the first role

27. * Independence: two events A and B are said to be independent if and only if either P(A|B)=P(A) or P(B|A)=P(B) - The fact that B has occurred has no effect on the probability that A will occur

28. Multiplicative Rule of probability P(AB)=P(A)P(B|A)=P(B)P(A|B) If A and B are independent events, P(AB)=P(A)P(B) Similarly, if A, B and C are mutually independent events, then P(ABC)=P(A)P(B)P(C)

29. Problem: baby’s gender a)Two couples plan to have child this year and what is the probability of both couples have boys? (Suppose having baby for each couple are independent events and there is an equal chance of having a boy or girl.) b)What is the probability that either couple or both has a boy? c)If you know one couple has had a boy what is the probability the other has had a boy.

30. How to Solve a Problem • Read the question and look for the phrase ‘what it is the probability of/that _____’ • Decide how to define the event(s) according to the blank • Decide if you are looking at an intersection (and), union (or), conditional (given), or just a probability • Take the given information and put in in a tree or a table, then fill in the blanks • Answer the question posed.

31. Tree • Multiply up the branches • Add down the nodes

32. *Probability table for the event A and B there is a two-way table whose four entries are the four intersection probabilities P(AB), P(A ), P( B ), P( ) and whose marginal row and column sums correspond to the unconditional probabilities P(A), P( ), P(B) and P( ) as in the table below: B A P(AB) P(A ) P(A) – add across P( B ) P( ) P( ) – add across P(B) P( ) sum down and across add down add down to equal 1

33. Problem: Is ABS useful? In a survey involving 100 cars, each vehicle was classifies according to whether or not it has antilock brakes system(ABS) and whether or not it has been involved in an accident in the past year. Suppose that any one of these cars is randomly selected for inspection. ABS No ABS Accident 3 12 No Accident 40 45 Transform this table into a probability table first.

34. Questions: • 1) What is probability that the car has been involved in an accident in the past year? • 2) What is probability that the car has ABS • 3) What is probability that the car has not been in an accident and has ABS • Given that the car has been involved in an accident, what is the probability that it has ABS? • Given that the car has ABS, what is the prob that it has been involved in an accident? • Given that the car does not have ABS, what is the prob that it has been involved in an accident?

35. Conditional probability and information update Consider this example first: Suppose 100 students live in Morrison Hall and assume at first that 50% of them love Chinese food and the other 50% don’t. If a person loves it, the probability that he order it is 80% for each meal. If the person doesn’t, the probability is 10%. Now, we observe an order of Chinese food has been delivered to the hall. Given this event, what is the probability the food was ordered by someone who loves Chinese food?

36. Bayes Rule Let S1, S2, S3,…Sk represent the k mutually exclusive, only possible states of nature with prior probabilities P(S1), P(S2),…P(Sk). If an event A occurs, the posterior probability of Si given A is the conditional probability

37. Bayes Rule • In English – The denominator has all possible ways of getting there while the numerator has the one way you want Chinese food example: P(L|O) = P(LO)/P(O) = P(O)P(O|L)/(P(O)P(O|L)+P(O)P(O|H))

38. Example 2 One part used in a production process is provided by three suppliers and supplier 1 provide 20%, 2 provide 30% and 3 provide 50%. The percentage of defective parts supplied by these three suppliers are 0.05, 0.02, and 0.01, respectively. If a part randomly selected from the production process is found to be defective, what is the probability that the part is provided by supplier 1?

39. Let’s Make a Deal • Monty Hall gives you the choice between 3 doors. One has a new car behind it and the other 2 have a goat. Once you have chosen a door Monty, knowing were the car is, reveals the contents of one of the doors you have not chosen (always the goat). He then gives you the option to choose one of the the other doors. Should you choose the other door?

40. Discrete Random variables and their probability distribution • Random variable: a variable is called a random variable if the value that it assumes has a probability of occurring. • 1) Random variable is a quantitative variable • 2) We can use values from the outcome of an experiment to get random variables.

41. Examples of Random Variables Toss a die x=number we observe x=number we observe-1 x=0 if observe odd number =1 if observe even Flip a coin x = 0 if H, x = 1 if T Select a student and measure their height x = height

42. Types of Random Variables * Discrete random variables have a countable number of values * Continuous random variables have infinite number of values Examples dice and coin toss – discrete height - continuous

43. Probability Distributions • Probability distribution for a discrete random variable is a formula, table, or graph that provides p(x), the probability associated with each of the value of x. Example 1: toss a die, x=number we observe formula table graph Example 2: toss two coins, what is the probability distribution of the the number of heads that we observe

44. Requirements for a discrete probability distribution • 0<=P(x)<=1 • Sum of P(x)=1 • Intuition: each simple event is assumed one and only one value. And each value of x corresponding to one or more than one simple events, but different x cannot correspond to same simple event. So sum of P(x) is sum of simple event probability.

45. Expected Value Expected value or population mean(mean) of random variable x with the probability distribution P(x) is given as Where the elements are summed over all values of the random variable x. Intuition:expected value is weighted average value of x

46. Examples of Expected Values • If you throw a dice what is the expected value? • If you flip two coins what is the expected value? • Should you (if you have rational expectations) mail (37 cent stamp) a response to Publishers Clearing House for a chance to win $10,000,000 when the odds of winning are one in 30,000,000?

47. Variance of a Random Variable The variance of random variable x with probability distribution P(x) and expected value E(x)= is given as Where the summation is over all values of the random variable x The Standard Deviation of random variable x is equal to the square root of its variance.

48. Example A random variable has the following probability distribution: Construct a probability histogram Find E(x), variance, and standard deviation What does Tchebysheff tell us about this distribution

49. Example from article in class • A recent survey showed that 57% of French people have an unfavorable view of America. You ask at random 2 French folks what there opinion of America was, call x the number of French who have an unfavorable view. • Find the probability distribution and construct the histogram. • What is the probability that at least one will dislike American