· Probability is always a fraction between 0 & 1

1. Probability·fraction that tells how likely something ` is to happen·the relative frequency that an event will occur

2. ·Notation: P(x) means the probability of event “x”. So P(heads) could mean the probability of getting heads when flipping a coin.

3. · Probability is always a fraction between 0 & 1

4. ·If P(x) = 0, then the event is impossible (can’t ever happen).· If P(x) = 1, then the event is certain (must always happen).

8. Probability can also be expressed as a percent (which is officially called chance)·Chance must always be between 0% and 100%.

9. Basic rule for finding probability:Desirable OutcomesP(x) = ------------------------------Total Outcomes

10. Ways to get what you wantP(x) = -------------------------------Ways to get anythingP(x) = x/n

11. EXAMPLE:One version of theMagic 8-Ball has 20different responses …9 “yes”8 “no”3 unresponsive

12. If you ask the Magic 8-Ball a question, what is the probability the answer is equivalent to “yes”?

13. 9 “yes”8 “no”3 unresponsiveSo 9/20 or .45

14. EXAMPLE:In a standard deck there are 52 cards.If you pick a card at random, what is the probability …

15. * It is a king?* It is a spade?* It is the queen of hearts?

16. * It is a king?4/52 or 1/13* It is a spade?13/52 or ¼ * It is the queen of hearts?1/52

17. EXAMPLE:When you flip a coin, what is the probability you get “heads”?

18. EXAMPLE:When you flip a coin, what is the probability you get “heads”? ½ or .5 or 50%

19. Theoretical Probability· Mathematical prediction of what the probability should be· Never changes

20. Empirical Probability· a.k.a. “Experimental Probability” or “Experiential Probability”· Actual relative frequency, based on past experience

21. Can change with circumstances· Rarely exactly equals the theoretical probability.

22. THEORETICAL what it should beEMPIRICAL what you actually got

23. Law of Large Numbers· If an event happens over and over again, the empirical probability will approach the theoretical probability

24. Law of Large Numbers

25. Sample Space· All the possible outcomes for an event

26. Examples:The sample space for rolling a die is { 1, 2, 3, 4, 5, 6 }The sample space for letter grades on a test is{ A, B, C, D, F }

27. The sum of all the probabilities in a sample space must equal 1 (because there is a 100% chance that something will happen).· So, Σ P(x) = 1

28. Complement· “everything but”· all the events in the sample space except the one in question· NOT some event

29. Notation: X’ (X-prime) or ~X (tilde-X)· P(x’) means the probability that an event won’t happen

30. ·P(x’) = 1 – P(x)·So if P(x) = ¼, then P(x’) = ¾.

31. EXAMPLE:One version of theMagic 8-Ball has 20different responses …9 “yes”8 “no”3 unresponsive

32. If you ask the Magic 8-Ball a question, what is the probability the answer is NOT “no”?

33. 9 “yes”8 “no”3 unresponsive(1 – 8/20) = 12/20 = .6

34. EXAMPLE:If you draw a card out of a deck of cards, what is the probability it is NOT a diamond?

35. EXAMPLE:If you draw a card out of a deck of cards, what is the probability it is NOT a diamond?1 – 13/52 = 39/52 or 3/4

36. EXAMPLE:In a poll conducted shortly after his inauguration, 45% of Americans said they supported President Trump. What percent did not support him?

37. EXAMPLE:In a poll conducted shortly after his inauguration, 45% of Americans said they supported President Trump. What percent did not support him? 100 – 45 = 55%

38. Fundamental Principle of Counting· If one event can happen in “x” ways and another event can happen in “y” ways, then the 2 events can happen together in x•y ways.

39. · When more than one thing happens at once, multiply to find the total possible outcomes.

40. EXAMPLEIf you roll two dice, how many ways could they land?

41. 6 x 6 = 36

42. EXAMPLEYou draw a card from a deck of cards, put it aside, and draw another card.How many ways can you do this?

43. 52 x 51 = 2652

44. EXAMPLEAs of 2013, most Iowa license plates have the format “ABC 123”. How many plates are possible with this format?

45. 26 x 26 x 26 x 10 x 10 x 1017,576,000

46. Mutually Exclusive·2 things that can’t happen at the same time

47. Independent· 2 events where the occurrence of one does not affect the occurrence of the other

48. Compound ProbabilityAND and OR

49. ANDP(A and B) = P(A)•P(B|A)·P(B|A) means the probability that B will occur, given that A has already occurred.

50. ·P(B|A) is read “the probability of B, given A”·You multiply probabilities in and problems.