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College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson

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Probability and Statistics 9

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Probability 9.2

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Overview In this section, We study probability, which is the mathematical study of “chance.”

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What is Probability?

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

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

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

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

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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.

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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-tossing experiment is S = {H, T}.

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

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Experiments with Equally Likely Outcomes We will be concerned only with experiments for which all the outcomes are equally likely. 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 about as many heads as tails will show up.

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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 ace from a deck of cards. Any particular set of outcomes is a subset of the sample space.

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

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E.g. 1—Events in a Sample Space An experiment consists of tossing a coin three times and recording the results in order. List the outcomes in the sample space, then list the outcome in each event. (a) The event E of getting “exactly two heads.” (b) The event F of getting “at least two heads.” (c) The event G of getting “no heads.”

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E.g. 1—Events in a Sample Space We write H for heads and T for tails. So the outcome HTH means that the three tosses resulted in Heads, Tails, Heads, in that order. The sample space is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

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E.g. 1—Events in a Sample Space The event E is the subset of the sample space S that consists of all outcomes with exactly two heads. Thus, E = {HHT, HTH, THH} Example (a)

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E.g. 1—Events in a Sample Space The event F is the subset of the sample space S that consists of all outcomes with at least two heads. Thus, F = {HHH, HHT, HTH, THH} Example (b)

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E.g. 1—Events in a Sample Space The event G is the subset of the sample space S that consists of all outcomes with no heads. Thus, G = {TTT} Example (c)

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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.

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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.

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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

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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 of an event is to 1, the more likely the event is to happen. The closer to 0, the less likely.

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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.

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E.g. 2—Finding the Probability of an Event A coin is tossed three times, and the results are recorded in order. Find the probability of the following. (a) The event E of getting “exactly two heads.” (b) The event F of getting “at least two heads.” (c) The event G of getting “no heads.”

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E.g. 2—Probability of an Event By the results of Example 1 the sample space S of this experiment contains 8 outcomes. The event E of getting “exactly two heads” contains 3 outcomes. So, by the definition of probability, Example (a)

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E.g. 2—Probability of an Event The event F of getting “at least two heads” has 4 outcomes. So, Example (b)

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E.g. 2—Probability of an Event The event G of getting “no heads” has one outcome. So, Example (c)

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Calculating Probability by Counting

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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 elements in these sets. The counting techniques that we learned in the preceding sections will be very useful here.

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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 cards are spades? The experiment here consists of choosing five cards from the deck. The sample space S consists of all possible five-card hands.

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

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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 spades is

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

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Understanding a Probability What does the answer to Example 3 tell us? Since 0.0005 = 1/2000, this means that if you play poker many, many times, on average you will be dealt a hand consisting of only spades about once every 2000 hands.

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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 from the bag, what is the probability that both are defective?

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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).

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

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The Complement of an Event

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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 E by E′.

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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

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Probability of the Complement of an Event Let S be the sample space of an experiment, and E and event. Then the probability of E′, the complement of E, is

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Probability of the Complement of an Event This is an extremely useful result. It is often difficult to calculate the probability of an event E. But, it is easy to find the probability of E′.

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E.g. 5—Finding a Probability Using 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 ball is red?

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Let E be the event that at least one red ball is drawn. It is tedious to count all the possible ways in which one or more of the balls drawn are red. So let’s consider E′, the complement of this event. E′ is the event that none of the balls drawn are red. E.g. 5—Finding a Probability Using the Complement of an Event

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

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By the formula for the complement of an event, we have E.g. 5—Finding a Probability Using the Complement of an Event

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The Union of Events

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If E and F are events, what is the probability that E or F occurs? The word or indicates that we want the probability of the union of these events. That is,.

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The Union of Events So, we need to find the number of elements in. If we simply add the number of element in E to the number of elements in F, then we would be counting the elements in the overlap twice. Once in E and once in F.

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The Union of Events So to get the correct total, we must subtract the number of elements in. Thus,

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The Union of Events Using the definition of probability, we get

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Probability of The Union of Events If E and F are events in a sample space S, then the probability of E or F is

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E.g. 6—Finding the Probability of the Union of Events What is the probability that a card drawn at random from a standard 52-card deck is either a face card or a spade? We let E and F denote the following events: E: The card is a face card. F: The card is a spade.

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E.g. 6—Finding the Probability of the Union of Events There are 12 face cards and 13 spades in a 51-card deck, so

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E.g. 6—Finding the Probability of the Union of Events Since 3 cards are simultaneously face cards and spades, we have

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E.g. 6—Finding the Probability of the Union of Events Thus, by the formula for the probability of the union of two events, we have

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Mutually Exclusive Events Two events that have no outcome in common are said to be mutually exclusive. This is illustrated in the figure.

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Probability of the Union of Mutually Exclusive Events If E and F are mutually exclusive events in a sample space S, then the probability of E or F is

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E.g. 7—Finding the Probability of the Union of Mutually Exclusive Events A card is drawn at random from a standard deck of 52 cards. What is the probability that the card is either a seven or a face card? Let E and F denote the following events: E:The card is a seven. F:The card is a face card.

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E.g. 7—Finding the Probability of the Union of Mutually Exclusive Events A card cannot be both a seven and a face card. Thus, the events are mutually exclusive.

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E.g. 7—Finding the Probability of the Union of Mutually Exclusive Events We want the probability of E or F. In other words, the probability of.

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E.g. 7—Finding the Probability of the Union of Mutually Exclusive Events By the formula,

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Conditional Probability and the Intersection of Events

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The Intersection of Events When we calculate probabilities, there sometimes is additional information that may alter the probability of an event. The probability of an event E given that another event F has occurred is expressed by writing P(E | F) = The probability of E given F

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The Intersection of Events Let E be the event of “getting a two,” and let F be the event of “getting an even number.” P(E | F) = P(The number is two given that the number is even)

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Conditional Probability Let E and F be events in a sample space S. The conditional probability of E given that F occurs is

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E.g. 8—Finding Conditional Probability A mathematics class consists of 30 students; 12 of them study French, 8 study German, 3 study both of these languages, and the rest do not study a foreign language. If a student is chosen at random from this class, find the probability of each of the following events.

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(a) The student studies French. (b) The student studies French, given that he or she studies German. (c) The student studies French, given that he or she studies a foreign language. E.g. 8—Finding Conditional Probability

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Let F = The student studies French G = The student studies German L = The student studies a foreign language

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E.g. 8—Finding Conditional Probability There are 30 students in the class, 12 of whom study French, so Example (a)

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E.g. 8—Finding Conditional Probability The probability that a student studies French given that the student studies German. Since eight students study German and three of these study French, it is clear that the required conditional probability is 3/8. Example (b)

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E.g. 8—Finding Conditional Probability The number of students who study a foreign language is 9 + 3 + 5 = 17. Example (c)

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Conditional Probability If we start with the expression for conditional probability and then divide numerator and denominator by n(S).

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Probability of the Intersection of Events If E and F are events in a sample space S, then the probability of E and F is

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E.g. 9—Finding the Probability of the Intersection of Events Two cards are drawn, without replacement, from a 52-card deck. Find the probability of the following events. (a)The first card drawn is an ace and the second is a king. (b)The first card drawn is an ace and the second is also an ace.

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E.g. 9—Probability of Intersection of Events Let E be the event “the first card is an ace,” and let F be the event “the second card is a king.” We are asked to find the probability of E and F. Now, P(E) = 4/52. After an ace is drawn, 51 cards remain in the deck; of these, 4 are kings, so P(F|E) = 4/51. Example (a)

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Let E be the event “the first card is an ace,” and let H be the event “the second card is an ace.” The probability that the first card drawn is an ace is P(E) = 4/52. After an ace is drawn, 51 cards remain; of these, 3 are aces, so P(H|E) = 3/51. E.g. 9—Probability of Intersection of Events Example (b)

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The Intersection of Independent Events When the occurrence of one event does not affect the probability of another event: We say that the events are independent. For instance, if a fair coin is tossed, the probability of showing heads on the second toss is 1/2. –This is regardless of the outcome of the first toss. –So, any two tosses of a coin are independent.

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Probability of the Intersection of Independent Events If E and F are independent events in a sample space S, then the probability of E and F is

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E.g. 10—Finding the Probability of Independent Events A jar contains five red balls and four black balls. A ball is drawn at random from the jar and then replaced. Then, another ball is picked. What is the probability that both balls are red?

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E.g. 10—Finding the Probability of Independent Events The events are independent. The probability that the first ball is red is 5/9. The probability that the second ball is red is also 5/9. Thus, the probability that both balls are red is

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