Probability

1. Probability 11.6 Learn the basic concepts about probability Calculate the probability of compound events Calculate the probability of independent and dependent events

2. Definition of Probability A result from an experiment is called an outcome. The set of all possible outcomes is the sample space. Any subset of a sample space is an event.

3. Definition of Probability If the outcomes of a finite sample space S are equally likely and if E is an event in S,then the probability of E is given by where n(E)and n(S) represent the number of outcomes in E and S, respectively.

4. Probabilities Since n(E) ≤n(S),the probability of an event E satisfies 0 ≤ P(E)≤ 1. IfP(E)= 1,then event E is certain to occur. If P(E)= 0,then event E is impossible.

5. One card is drawn at random from a standard deck of 52 cards. Find the probability thatthe card is an ace. Solution Sample space S consists of 52 outcomes, the 52 cards in the deck. n(S) = 52 Each outcome is equally likely. Let E represent the event of drawing an ace. There are 4 aces, n(E) = 4. Example: Drawing a card

6. In 2012, there were 114,448 patients waiting for an organ transplant. The table lists the number of patients waiting for the most common type of transplants. None of these people need two or more transplants. Approximate the probability that a transplant patient chosen at randomwill need a) kidney or a heart b) neither a kidney nor a heart. Example: Estimating probability of organ transplants

7. a) E = kidney or heart = 92,346 + 3186 = 95,532 In 2012, about 83% of transplant patients needed either a kidney or a heart. Example: Estimating probability of organ transplants

8. b) F = other than a kidney or heart = 114,448 – 95,532 = 18,916 In 2012, about 17% of transplant patients needed an organ other than a kidney or a heart. Example: Estimating probability of organ transplants

9. Complement of an Event Notice that P(E)+ P(F)= 1, The events E and F are complementsbecause where denotes the empty set. That is, a transplantpatient is either waiting for a kidney or a heart (event E) or not waiting for a kidney or aheart (event F). The complement of E may be denoted by E′.

10. Probabilities and Venn Diagram The sample space S of an experiment is the union of the disjoint sets E and E′.

11. Probability of a Complement Let E be an event and E′ be its complement. If the probability of E is P(E),then theprobability of its complement is given by

12. Compound Events Frequently the probability of more than one event is needed. For example, suppose a college with a total of 2500 students has 225 students enrolled in college algebra, 75 intrigonometry, and 30 in both.

13. Compound Events Let E1 denote the event that a student is enrolled in collegealgebra and E2 the event that a student is enrolled in trigonometry. Then the Venn diagramdescribes thesituation.

14. Compound Events In this Venn diagram, it is important that the 30 students taking both courses not becounted twice. Set E1 has a total of 195 + 30 = 225students, and set E2contains 45 + 30 = 75students.

15. Probability of the Union of Two Events For any two events E1and E2,

16. Suppose two dice are rolled. Find the probability that the dice show either a sum of eightor a pair. Solution Example: Rolling dice

17. Because each die can show six different outcomes, there are a total of 6 •6 = 36outcomes in the sample space S.Let E1 denote the event of rolling a sum of eight and E2 theevent of rolling a pair. Then E1 = {(6, 2), (5, 3), (4, 4), (3, 5), (2, 6)} and E2 = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} E1UE2 = {(4, 4)} Example: Rolling dice

18. Since n(S) = 36, n(E1) = 5, n(E2) = 6, and n(E1) = 1, the following can be computed. Example: Rolling dice

19. Mutually Exclusive Events If , then the events E1and E2are mutually exclusive. Mutually exclusive events have no outcomes in common, soP(E1E2) = 0. In this case, P(E1 E2) = P(E1) P(E2) .

20. Find the probability of drawing either an ace or a king from a standard deck of 52 cards. Solution The event of drawing an ace and drawing a king are mutually exclusive. No card can be an ace and a king. Example: Drawing cards

21. Independent and Dependent Events Two events are independent if they do not influence each other. Otherwise they are dependent. An example of independent events would be one coin being tossed twice. The result of the first toss does not affect the second toss.

22. Probability of Independent Events If E1and E2are independent events, then P(E1E2) = P(E1) • P(E2) .

23. Suppose a coin is tossed twice. Determine the probability that the result is two heads. Solution E1 = event of a head on the first toss E2 = event of a head on the second toss The two events are independent. Example: Tossing a coin

24. Probability of Dependent Events If E1and E2are dependent events, then P(E1E2) = P(E1) • P(E2, given that E1 occurred).

25. Find the probability of drawing two hearts from a standard deck of 52 cards, when the first card isa) replaced before drawing the second card; b) not replaced. Solution a) E1 = heart, E2 = heart; independent events Example: Drawing cards

26. b) The first card is not replaced, the outcome of the second card is influenced by the first card. The events are dependent. Example: Drawing cards

27. The table shows the number of students (by gender) registered for either a Spanish or a French class. No student is taking both languages. Example: Calculating the probability of dependent events

28. If one student is selected at random, calculate each of the following. a) The probability that the student is female. Example: Calculating the probability of dependent events Solution

29. If one student is selected at random, calculate each of the following. b) The probability that the student is taking Spanish, given that the student is female. Example: Calculating the probability of dependent events Solution

30. If one student is selected at random, calculate each of the following. c) The probability that the student is female and taking Spanish. Example: Calculating the probability of dependent events Solution