Probability

1. Probability

2. THE BASIC LAW OF PROBABILITY • ALL OUTCOMES BEING LIKELY, AN EVENT’S PROBABILITY = • THE WAYS OF THE EVENT YOU’RE INTERESTED IN OCCURING/TOTAL POSSIBLE NUMBER OF OUTCOMES (n) • Written as a formula, this would be: • P(A)=number of events in A / total number of trials n

3. Some basic probabilities • What is the probability that you roll a three on a six sided die? • What is the probability of drawing the 10 of hearts from a deck of cards? • What is the probability that you flip a heads? • These are easy, what about when we have to deal with events that occur in several stages?

4. We make trees

5. Example • You flip a coin 3 times, what is the probability of • 3 heads? • Heads, Tails, Heads • At least one Tails • 2 or more heads?

6. Make your tree • Each new section represents a trial

7. A handy formula for finding n • TO FIND n(THE TOTAL POSSIBLE OUTCOMES) • HANDY FORMULA: • Number of possible outcomes per object ^ number of objects • 4 sides of each COIN ^ 3 individual COINS • This is your n=2 ^ 3 = 8

8. EXAMPLE • YOU ROLL A PAIR OF 4-SIDED DICE. EACH OUTCOME HAS A PROBABILITY OF ______________

9. HOW TO SOLVE IT: • FIRST, FIND N (THE TOTAL POSSIBLE OUTCOMES) • HANDY FORMULA: • Number of possible outcomes per object ^ number of objects • 4 sides of each die ^ 2 individual dice • This is your n • THE PROBABILITY FOR EACH OUTCOME IS 1/N OR 1/16 • THIS COULD ALSO BE SHOWN BY A TREE

10. Probability tree

11. EXAMPLE • YOU ROLL A PAIR OF 4-SIDED DICE. WHAT IS THE PROBABILITY THAT THE SUM IS EVEN?

12. HOW TO SOLVE IT: • We already made the tree: • Just go through it, find the even sums, and divide by 16

13. EXAMPLE • YOU ROLL A PAIR OF 4-SIDED DICE. WHAT IS THE PROBABILITY THAT THE FIRST ROLL IS BIGGER THAN YOUR SECOND ROLL?

14. HOW TO SOLVE IT: • USE THE TREE

15. Rules of Probability Unions, Intersections

16. A trick to remember the difference Intersections: • Intersection: MIT DUSP is located at the intersection of Mass Ave AND Vasser. • An intersection contains the elements in A AND B • Example: You have two sets • A={2,4,6,8,10} B={1,2,3,4,5} • What is A B?

17. Unions A trick to remember the difference • Union: Think of a union as a marriage between two sets: When people get married they bring their belongings into one house. Items which either he OR she owned are now in the new house. • A Union contains elements in A OR in B • Example: A={2,4,6,8,10} B={1,2,3,4,5} • What it A B? • The number is in A or B

18. The General Rule for addition • Used for unions of probabilities • What is the probability that either A or B happens? • The formula is P(A U B) = P(A) + P(B) – P(A∩B)

19. The probability of 3 events occurring (A and B and C) • P( A U B U C)= P(A) + P(B) + P(C) – P(A B) –P(A C)-P(B C) ∩ ∩ ∩

20. Example Pi alpha member Non Member • You qualify to be in the English civil service if you have a degree, you are a member of Pi Alpha, or you pass an exam. What is the probability a person is on the list because they passed an exam, had a degree, or was a member of Pi Alpha?

21. Add the probabilities up… • This question is asking you to calculate an EITHER probability, so you use the addition rule • It is asking for three events, so you need to add all three, subtract shared, and then re-add the overlap of all three • Find probability of Passing: 120/250 • Find the probability of Having a degree: 130/250 • Being a member: 60/250 Pi alpha member Non Member

22. Last steps: • The probability of passing, having a degree, or being a member is: 0.48+0.52+0.24 • The answer is 1.24 (which we know can’t be right) • We now need to subtract P(A and B) P(A and C) and P(B and C) • Go back to the tables to get these numbers • This is • P(member and Pass) =(40/250)=0.16 • P(member and degree) =(30/250)=0.12 • P(Pass and degree)= (90/250)=0.36

23. Last steps: • Then re-add P(A and B and C) • The table says it is 26/250=0.104 So: • 1.24-(0.16+0.36+0.12) + 0.104= • 0.704 is the probability a person is on the list because they passed an exam, had a degree, or was a member of Pi Alpha?

24. General multiplication rule • Used when you want to find the joint probability of two events • This is an “and” probability • The probability of A and B, or P(A B) • Equals P(A) * P(B | A) • If A and B are independent, you can just take P(A) * P(B) • This is super simple and is best illustrated with an example ∩

25. Independence • When the probability of an event is not influenced by the event before it

26. Independence • A jar has 3 red marbles, 3 blue marbles, and 2 yellow marbles. • You pick out one marble. What is the probability it is red? • It was red. What is the probability that your next one is red, if you don’t put the red back in the jar? • What if you put it back in and then pick?

27. Example : conditional probability and independence • You have 4 females and 2 males in a group. You need to select two people to be on a committee. You want to choose at random, and you can’t choose the same person twice. What is the probability of…. • (F F) • M F) ∩ ∩

28. Make a tree

29. The formula for what you just did • P(M F) = P(M) * P(F | M) • P(F F)=P(F) * P(F | F) • This is where the conditional is important, if you choose a female first, the total number of females that can be selected from decreased ∩ ∩

30. Are these events independent? • Use this formula for independence: If A and B are independent then P(B)= P(B|A) and P(A)=P(A|B) • In this example, that means that the probability of choosing a male is equal to the probability that you choose a male given you chose a female first. • It also means the probability of choosing a female equals the probability that you chose a female given you chose a male first

31. Example General Multiplication Rule • You are taking pizza orders. A customer can order a small, medium, or large. They can choose thin or thick crust. They can choose up to two toppings, peperoni or mushrooms. • Are these likelihoods independent? • This is a real life example of conditional probability (several conditions across several stages)

32. Sample Problem: General Rule of multiplication: • What is P(Small Thin No peperoni mushrooms)? • What is the probability of small ∩ thick crust? • What is the probability of small ∩ thick crust ∩ peperoni? ∩ ∩ ∩

35. Examples where you’d use both Multiplication and Addition rules • Often, in probability you don’t use one rule on its own • Trees help you determine when and where to use each rule • When you make a tree and move left to right, it is the multiplication rule • When you are going up and down, it is the addition rule. • Let’s show this with an example.

36. Sample problem 3 • There are three restaurants in town. They get 50%, 30%, and 20% of the business. You know that 70% of the customers that leave Restaurant 1 are satisfied. 60% at Restaurant 2 are satisfied and 50% that leave restaurant 3 are satisfied. What is the probability that someone eating in this town leaves satisfied?

37. Make your tree

38. Multiply through each branch to get the conditional probabilities

39. Add down the line to get total p for satisfaction • You’re first multiplying across to get the conditionals • Then you’re adding up and down, to get the probability of being satisfied at 1, 2, OR 3 • .35+.18+.1=.63 • You have a 63% change of being satisfied when eating out in town • The formula for what you just did: • What is the P[(eat at R1 satisfies) (eat at R2 satisfied) eat at R 3 satisfied] ∩ ∩ ∪ ∪ ∩

40. One last example • You’re playing in a chess tournament. Your probability of winning against half of the players (type 1) is 0.3. Your probability of winning against a quarter of the players (type 2) is 0.4, and it is .5 when playing against the other quarter of the players (type 3). • What’s the probability of winning when a random opponent is chosen?

41. Make your table

42. Multiply out your conditionals • P(win|type 1), P(win|type 2), and P(win|type3) • They are independent so you can just multiply through L to R

43. Sum according to addition rule • 0.15+0.1+0.125=0.375 • Your chance of winning given a randomly chosen opponent is 0.375 or just over 37% • The formula shows why you’d conclude with the addition rule: • P(win|type 1) (win|type2) (win|type 3) ∪ ∪

44. Conditional Probability • “Given X, what is the probability of Y” • Example: You’re picking one person at random from the class. Given the person in the class is a female, what the the probability he or she is blonde? • What statisticians would write: P(Blonde | Female) • Tips: your total (n, or the number you divide by is only the girls! Not the whole class) • (# of blondes/#of girls)

45. From your pset! • Given your cloth is from a hand loom, what is the probability • that the quality is poor? • Locate Handloom cloth • How many total pieces are there made by a hand loom? • How many of those are of poor quality?

46. Hints for solving Probability word problems • When there is already a table, diagramming a tree is unnecessary • Be careful to take the right total (n, denominator) • Especially in conditional probabilities! • The simplest example of a conditional probability is the blonde | woman example we did above, store that in your head for easy reference, and so you’re not intimidated by the “ | “

47. The Normal Distribution

48. Probability Distributions A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence.

49. Real life normal distribution

50. Distributions fit with different types of variables: Discrete variables: takes on a countable number of values      -the number of job classifications in an agency      -the number of employees in a department       -the number of training sessions  Continuous variables: takes a countless (or super big) range of numerical values       -temperature      -pressure      -height, weight, time      -Dollars: budgets, income. (not strictly continuous) but they can take so many values that are so close that you may as well treat them that way