Probability

1. Probability

2. Sets A set is a collection of objects. These objects could be discrete singletons (such as the integers), or could be points on a continuum (such as the points on the real line). Probability is a function defined on sets.

3. Examples:

4. The intersection of two sets is the set of elements in both sets. The intersection operation is denoted by the symbol . The intersection operation is like a logic AND. The union of two sets is the set of elements in either sets. The union operation is denoted by the symbol . The union operation is like a logic OR. Two sets are said to be mutually exclusive if their intersection is the null set. (There is no element in both sets.)

5. It is possible to re-partition any collection of subsets into a collection of mutually-exclusive subsets. Consider two (overlapping) sets A and B. A B

6. While A and B are not mutually exclusive, the set A - AB is mutually exclusive of B. A - AB AB B

7. The complement of a set are those elements notin the given set. Given a set A, the complement of the set is denoted by AC. A AC

8. We can combine the operations of union, intersection and complementation. Given two sets, A and B, let us first take the union of the two sets.

9. A B A B

10. Let us see if we can relate the union to the complements of the two sets.

11. AC A

12. BC B

13. (A B)C = AC BC A B

14. Exercise: Draw a Venn diagram exemplifying (A B)C = AC BC

15. We can take the union or intersection of a sequence of sets. Suppose we have n sets, Ai (1  i  n ) such that Ai  Aj if i  j, then

16. Suppose we have n sets, Ai (1  i  n ) such that Ai  Aj if i  j, then We can take limits of such sequences.

17. Ai  Aj Ai  Aj

18. Suppose we have neither Ai  Aj or Ai  Aj , then the limit may not exist. Even though the limit does not exist the lim inf and the lim sup do exist:

19. Probability is a measure of the likelihood of some event occurring. An event is a subset of the set of all possible outcomes. As an example, when we roll a die, the set of all possible outcomes is {1,2,3,4,5,6}. The possible events are {1}, {2}, {3}, {4}, {5}, {6}, {even}, {odd}, ….

20. The set of all possible outcomes is denoted by W. An event is denoted by various symbols, A, B, … . For each event, A, we must have A  W . The probability measure can be applied to any event. In the die example W = {1,2,3,4,5,6}. The possible events are {1}, {2}, {3}, {4}, {5}, {6}, {1, 3 or 5}, {2, 4 or 6}}, …. The probability of any singleton event is 1/6.

21. Definition Theprobability of an event is a ratio: where #E is the cardinalityor number of elements of set E and Wis the universal set.

22. Example: Suppose that we roll a die. Find the probability that (1) a two shows up, (2) a six shows up, (3) a two or a six shows up, (4) an even number of dots shows up (5) an even number of dots show up or the number of dots is less than three Let x be the number of dots which show-up when we roll the die.

24. This probability was determined by dividing the two sets x<3 and x is even into two mutually exclusive sets.

25. If two events, A and B, are mutually exclusive, then If n events, {Ai} are mutually exclusive, then

26. If two events, A and B, are such that A  B then If n events, {Ai} are such that Ai  Aj if i  j, then

27. Example: Roll a pairof dice. Let x be the total number of dots which show up or the sum of the number of dots on both dice. The number of combinations of dots for two dice is equal to six squared (62): for each possible number of dots for the first die, we have six possible number of dots for the second die.

33. Calculating the probability ratio is relatively easy. The hardest part of doing probability calculations is determining the number of occurrences of an event.

34. Multiplication Principle, Permutations and Combinations How many possibilities are there in a sequence of multiple events?

35. Multiplication Principle Suppose that we performed an event, say, flipping a coin, k times. What are the number of different ways that these k events can happen? For k=2, the number of different ways that both events can occur can be found by finding all possible pairs of events (e.g., “heads-heads,” “heads-tails,” “tails-heads,” “tails-tails”).

36. Heads Heads, Heads Heads, Tails Tails Tails, Heads Tails, Tails First Flip Second Flip

37. If we flip three coins, the diagram is as follows:

38. If two events, A and B, are independent such that the occurrence of A has no influence upon the occurrence (or non-occurrence) of B, then If n events, {Ai} are independent, then

39. Example: Find the number of different ways in which you can flip a coin, roll a die and flip a coin? Answer: by drawing a tree (either physically, or in our heads), we see that the answer is (2)(6)(2) = 24.

40. Replacement In our previous examples, one event did not affect subsequent events. Such is not always the case. If we are dealing cards from a deck, there are fifty-two different ways to draw the first card, but only fifty-one ways to draw the second card. It would be possible for there to be fifty-two ways of drawing both cards, if the first card were replaced in the deck before the second card were drawn.

41. Example: How many different ways can we draw five cards? Answer: There are fifty-two possibilities for the first card, fifty-one possibilities for the second card (assuming the first card was not replaced), etc. So the final answer is (52)(51)(50)(49)(48).

42. Permutations In the previous example, the answer was equal to (n)(n-1)…(n-k+1), where n=52 and k=5. It would be nice if there were a more compact way of representing this operation.

43. In the previous example (n=52 and k=5), The notation that we use for the number of permutations is

44. Example: How many different ways can you deal a hand of bridge (with thirteen cards). Answer: Using our permutation notation, .

45. Example: Suppose we had a box with n slots, and we wish to put k < nmarbles into kof the nslots. No more than one marble will go into any one slot. … M M M M 1 2 3 k …… 1 2 3 n

46. This problem can be solved using the multiplication principle. The result is the same as drawing kcards from a deck of n cards: there are n possibilities for the first marble, n-1 possibilities for the second marble, etc. … M M M 1 2 3 k M …… n-1 left 1 2 3 n

47. … M M 1 2 3 k M M …… n-2 left 1 2 3 n

48. Specific Example: Suppose we had four slots and two marbles. The two marbles are red (R) and blue (B). Show all possible ways in which the two marbles can be placed in the four slots. R B R B R B R B R B R B B R B R B R B R B R B R

49. Notice that in the previous example, n=4 and k=2, so the number of permutations is (correctly)

50. Combinations Suppose in our previous example (nslots and kmarbles), that all of the marbles were the same color. The number of possibilities would not be as great. As an example, would be the same as R B B R The number of possibilities would be reduced by a factor equal to the number of ways kmarbles can be rearranged in k slots. This factor is simply k!