College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Counting and Probability 10

3. Probability 10.3

4. Overview • In the preceding chapters, we modeled real-world situations. • These were modeled using precise rules, such equations or functions.

5. Overview • However, many of our everyday activities are not governed by precise rules. • Rather, they involve randomness and uncertainty.

6. Overview • How can we model such situations? • Also, how can we find reliablepatterns in random events? • In this section, we will see howthe ideas of probability provideanswers to these questions.

7. Rolling a Die • Let’s look at a simple example. • We roll a die, and we’re hoping to geta “two”. • Of course, it’s impossible to predict whatnumber will show up.

8. Rolling a Die • But, here’s the key idea: • We roll the die many many times. • Then, the number two will show upabout one-sixth of the time.

9. Rolling a Die • This is because each of the six numbers is equally likely to show up. • So, the “two” will show up abouta sixth of the time. • If you try this experiment, you willsee that it actually works!

10. Rolling a Die • We say that the probability (or chance) of getting “two” is 1/6.

11. Picking a Card • If we pick a card from a 52-card deck, what are the chances that it is an ace? • Again, each card is equally likely to be picked. • Since there are four aces, the probability (or chances) of picking an ace is 4/52.

12. Probability and Science • Probability plays a key role in many sciences. • A remarkable example of the use of probability is Gregor Mendel’s discovery of genes. • He could not see the genes. • His discovery was due to applying probabilistic reasoning to the patterns he saw in inherited traits.

13. Probability • Today, probability is an indispensable tool for decision making. • It is used in business, industry, government, and scientific research. • For example, probability is used to • Determine the effectiveness of new medicine • Assess fair prices for insurance policies • Gauge public opinion on a topic (without interviewing everyone)

14. Probability • In the remaining sections of this chapter, we will see how some of these applications are possible.

15. What is Probability?

16. Terminology • To discuss probability, let’s begin by defining some terms. • An experiment is a process, such as tossing a coin or rolling a die. • The experiment gives definite results called the outcomes of the experiment. • For tossing a coin, the possible outcomes are “heads” and “tails” • For rolling a die, the outcomes are 1, 2, 3, 4, 5, and 6.

17. Terminology • The sample space of an experiment is the set of all possible outcomes. • If we let H stand for heads and T for tails,then the sample space of the coin-tossingexperiment is S = {H, T}.

18. Sample Space • The table lists some experiments and the corresponding sample spaces.

19. Experiments with Equally Likely Outcomes • We will be concerned only with experiments for which all the outcomes are equally likely. • We already have an intuitive feeling for whatthis means. • When we toss a perfectly balanced coin,heads and tails are equally likely outcomes. • This is in the sense, that if this experiment is repeated many times, we expect that aboutas many heads as tails will show up.

20. Experiments and Outcomes • In any given experiment, we are often concerned with a particular set of outcomes. • We might be interested in a die showing an even number. • Or, we might be interested in picking an acefrom a deck of cards. • Any particular set of outcomes is a subset of the sample space.

21. An Event—Definition • This leads to the following definition. • If S is the sample space of an experiment,then an event is any subset of the samplespace.

22. E.g. 1—Events in a Sample Space • An experiment consists of tossing a coin three times and recording the results in order. • The sample space is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

23. E.g. 1—Events in a Sample Space • The event E of showing “exactly two heads” is the subset of S. • E consists of all outcomes with two heads. • Thus, E = {HHT, HTH, THH}

24. E.g. 1—Events in a Sample Space • The event F of showing “at least two heads” is • F = {HHH, HHT, HTH, THH} • And, the event of showing “no heads” is G = {TTT}

25. Intuitive Notion of Probability • We are now ready to define the notion of probability. • Intuitively, we know that rolling a die may result in any of six equally likely outcomes. • So, the chance of any particular outcome occurring is 1/6.

26. Intuitive Notion of Probability • What is the chance of showing an even number? • Of the six equally likely outcomes possible, three are even numbers. • So it is reasonable to say that the chance of showing an even number is 3/6 = 1/2. • This reasoning is the intuitive basis for the following definition of probability.

27. Probability—Definition • Let S be the sample space of an experiment in which all outcomes are equally likely. • Let E be an event. • The probability of E, written P(E), is

28. Values of a Probability • Notice that 0 ≤ n(E) ≤ n(S). • So, the probability P(E) of an event is a number between 0 and 1. • That is, 0 ≤ P(E) ≤ 1. • The closer the probability is to 1, the more likely the event is to happen. • The close to 0, the less likely.

29. Values of a Probability • If P(E) = 1, then E is called the certain event. • If P(E) = 0, then E is called the impossible event.

30. E.g. 2—Finding the Probability of an Event • A coin is tossed three times, and the results are recorded. • What is the probability of getting exactlytwo heads? • At least two heads? • No heads?

31. E.g. 2—Finding the Probability of an Event • By the results of Example 1, the sample space S of this experiment contains eight outcomes. • The event E of getting “exactly two heads”contains three outcomes. • They are {HHT, HTH, THH}. • So, by the definition of probability,

32. E.g. 2—Finding the Probability of an Event • Similarly, the event F of getting “at least two heads” has four outcomes. • They are {HHH, HHT, HTH, THH}. • So,

33. E.g. 2—Finding the Probability of an Event • The event G of getting “no heads” has one element, so

34. Calculating Probabilityby Counting

35. Calculating Probability by Counting • To find the probability of an event: • We do not need to list all the elements in the sample space and the event. • What we do need is the number of elementsin these sets. • The counting techniques that we learned inthe preceding sections will be very usefulhere.

36. E.g. 3—Finding the Probability of an Event • A five-card poker hand is drawn from a standard 52-card deck. • What is the probability that all five cardsare spades? • The experiment here consists of choosingfive cards from the deck. • The sample space S consists of all possiblefive-card hands.

37. E.g. 3—Finding the Probability of an Event • Thus, the number of elements in the sample space is

38. E.g. 3—Finding the Probability of an Event • The event E that we are interested in consists of choosing five spades. • Since the deck contains only 13 spades,the number of ways of choosing five spadesis

39. E.g. 3—Finding the Probability of an Event • Thus, the probability of drawing five spades is

40. Understanding a Probability • What does the answer to Example 3 tell us? • Since 0.0005 = 1/2000, this means that if youplay poker many, many times, on averageyou will be dealt a hand consisting of onlyspades about once every 2000 hands.

41. E.g. 4—Finding the Probability of an Event • A bag contains 20 tennis balls. • Four of the balls are defective. • If two balls are selected at random fromthe bag, what is the probability that bothare defective?

42. E.g. 4—Finding the Probability of an Event • The experiment consists of choosing two balls from 20. • So, the number of elements in the sample space S is C(20, 2). • Since there are four defective balls,the number of ways of picking two defective balls is C(4, 2).

43. E.g. 4—Finding the Probability of an Event • Thus, the probability of the event E of picking two defective balls is

44. Complement of an Event • The complement of an event E is the set of outcomes in the sample space that is not in E. • We denote the complement of an event Eby E′.

45. Complement of an Event • We can calculate the probability of E′ using the definition and the fact that n(E′) = n(S) – n(E) • So, we have

46. Probability of the Complement of an Event • Let S be the sample space of an experiment, and E and event. • Then

47. Probability of the Complement of an Event • This is an extremely useful result. • It is often difficult to calculate the probabilityof an event E. • But, it is easy to find the probability of E′. • From this, P(E) can be calculated immediatelyby using this formula.

48. E.g. 5—The Probability of the Complement of an Event • An urn contains 10 red balls and 15 blue balls. • Six balls are drawn at random from the urn. • What is the probability that at least one ballis red?

49. E.g. 5—The Probability of the Complement of an Event • Let E be the event that at least one red ball is drawn. • It is tedious to count all the possible waysin which one or more of the balls drawnare red. • So let’s consider E′, the complement of thisevent. • E′ is the event that none of the balls drawnare red.

50. E.g. 5—The Probability of the Complement of an Event • The number of ways of choosing 6 blue balls from the 15 balls is C(15, 6). • The number of ways of choosing 6 ballsfrom the 25 ball is C(25, 6). • Thus,