Probabilities of Disjoint and overlapping events

1. Probabilities of Disjoint and overlapping events

2. Unions and Intersections • When you consider all the outcomes for either of two events, A and B, you form the union of A and B. • When you consider only the outcomes shared by both A and B, you form the intersection of A and B. • The union or intersection of two events is called the compound event.

3. Union B A

4. Intersection B A

5. Intersection of A and B is empty A B

6. Compound Events • To find P(A and B) you must consider what outcomes, if any, and in the intersection of A and B. • Two events are overlapping if they have one or more outcomes in common as seen in the UNION diagram. • Two events are disjoint,or mutually exclusive, if they have no outcomes in common, as shown in the 3rd diagram.

7. Probability of Compound Events • If A and B are any two events, then the probability of A or B is: P(A or B) = P(A) + P(B) – P(A and B) • If A and B are disjoint events, then the probability of A or B is: P(A or B) = P(A) + P(B)

8. Find probability of disjoint events • A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a 10 or a face card?

9. Find probability of disjoint events • Let event A be selecting a 10 and event B be selecting a face card. • A has 4 outcomes and B has 12 outcomes. Because A and B are disjoint, the probability is: P(A or B) = P(A) + P(B) =

10. Find probability of compound events • A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a face card or a spade?

11. Find probability of compound events • Let event A be selecting a face card and event B be selecting a spade. The events are shown with the overlapping events. B A 10 9 8 7 6 5 4 3 2 A K Q J K Q J K Q J K Q J

12. Find the probability of compound events • Remember: • P(A or B) = P(A) + P(B) – P(A and B) Thus the probability of drawing a spade or a face card is: P(A or B) =

13. Use the formula to find P(A and B) • Out of 200 students in a senior class, 113 students are either varsity athletes oron the honor roll. There are 74 seniors who are varsity athletes and 51 seniors who are on the honor roll. • What is the probability that a randomly selected senior is both a varsity athlete and on the honor roll?

14. Use a formula to find P(A and B) • Let event A be selecting a senior who is a varsity athlete and event B be selecting a senior on the honor roll. • From the given information you know: • P(A)= P(B)= P(A or B)= • Find P(A and B).

15. Use a formula to find P(A and B) • P(A or B) = P(A) + P(B) – P(A and B) • P(A and B) =

16. Practice • A card is randomly selected from a standard deck of 52 cards. Find the probability of the given event. • Selecting an ace or an eight • Selecting a 10 or a diamond

17. Practice Answers • Selecting an ace or an eight • Selecting a 10 or a diamond

18. Complements • The event A’, called the complement of event A, consists of all outcomes that are not in A. • The notations A’ is read “A prime or A complement” • The book uses the notation Āfor the complement and is read “A bar”. • A’ = Ā

19. Probability of the Complement of an Event • The probability of the complement of A is: P(A’) = 1 – P(A) or P(Ā) = 1 – P(A)

20. Find probabilities of complements • When two six-sided dice are rolled, there are 36 possible outcomes as shown in the table.

21. Find probabilities of complements • Find the probability of the given event: • The sum is not 6 • The sum is less than or equal to 9

22. Find probabilities of complements • The sum is not 6 • P(sum is not 6) = 1 – P(sum is 6) • The sum is less than or equal to 9: • P(sum ≤ 9) = 1 – P(sum > 9)

23. Use a complement in real life • A restaurant gives a free fortune cookie to every guest. The restaurant claims there are 500 different messages hidden inside the fortune cookies. • What is the probability that a group of 5 people receive at least 2 fortune cookies with the same message inside?

24. Use a complement in real life • The number of ways to give messages to the 5 people is 5005. The number of ways to give different messages to 5 people is 500  499 498 497 496. • So, the probability that at least 2 of the 5 people have the same message is: P(at least 2 are the same) = 1 – P(none are the same

25. Practice • Find P(A’) • P(A) = 0.45 • P(A) = ¼ • P(A) = 1 • P(A) = 0.03

26. Practice Answers • P(A’) = 0.55 • P(A’) = 3/4 • P(A’) = 0 • P(A’) = 0.97

27. Probabilities of independent and dependent events

28. Independent Events • Two events are independent if the occurrence of one has no effect on the occurrence of the other. • For instance, if a coin is tossed twice, the outcome of the first toss (heads or tails) has no effect on the outcome of the second toss.

29. Probability of Independent Events • If A and B are independent events, then the probability that both A and B occur is: P(A and B) = P(A)P(B) • More generally, the probability that n independent events occur is the product of the n probabilities of the individual events.

30. Probability of Independent Events • For a fundraiser, a class sells 150 raffle tickets for a mall gift certificate and 200 raffle tickets for a booklet of movie passes. You buy 5 raffle tickets for each prize. • What is the probability that you win both prizes?

31. Probability of Independent Events • Let events A and B be getting the winning ticket for the gift certificate and movie passes, respectively. The events are independent. So, the probability is: P(A and B) = P(A)P(B) =

32. Find the probability of 3 independent events • In a BMX meet, each heat consists of 8 competitors who are randomly assigned lanes from 1 to 8. • What is the probability that a racer will draw lane 8 in the 3 heats in which the racer participates?

33. Find the probability of 3 independent events • Let events A, B, and C be drawing lane 8 in the first, second, and third heats, respectively. The 3 events are independent. So, the probability is: P(A and B and C) = P(A)  P(B) P(C)

34. Use a complement to find a probability • While you are riding to school, your portable CD player randomly plays 4 different songs from a CD with 16 songs on it. • What is the probability that you will hear your favorite song on the CD at least once during the week (5 days)?

35. Use a complement to find a probability • For one day, the probability of not hearing you favorite song is: P(not hearing song) =

36. Use a complement to find a probability • Hearing or not hearing your favorite song on Monday, on Tuesday, and so on are independent events. So, the probability of hearing the song at least once is: P(hearing song) = 1 – [P(not hearing song)]5 =

37. Dependent Events • Two events are A and B are dependent events if the occurrence of one affects the occurrence of the other. • The probability that B will occur given that A has occurred is called the conditional probability of B given A and is written as P(B|A).

38. Probability of Dependent Events • If A and B are dependent events, then the probability that both A and B occur is: P(A and B) = P(A) P (B|A)

39. Find a conditional probability • The table shows the numbers of tropical cyclones that formed during the hurricane seasons from 1988 to 2004. • Use the table on the next slide to estimate: • The probability that a future tropical cyclone is a hurricane • The probability that a future tropical cyclone in the Northern Hemisphere is a hurricane.

40. Find a conditional probability # of hurricanes # of hurricanes in Northern Hemisphere Total # of cyclones Total # of cyclones in Northern Hemisphere • P(hurricane) = • P(hurricane | Northern Hemisphere) =

41. Comparing independent and dependent events • You randomly select two cards from a standard deck of 52 cards. • What is the probability that the first card is not a heart and the second card is a heart? • Find the probability first WITH REPLACEMENT, then WITHOUT REPLACEMENT.

42. Comparing independent and dependent events • Let A be “the first card is not a heart” and B be “the second card is a heart”. • If you replace the first card before selecting the second card, then A and B are independent events. • So, the probability is: P(A and B) = P(A)P(B)

43. Comparing independent and dependent events • If you do not replace the first card before selecting the second card, the A and B are dependent events. • So, the probability is: P(A and B) = P(A)  P (B|A)