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Probability and Sample Spaces

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EXAMPLE 17

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Let s Make a Deal

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On the television program Let s Make a Deal, a contestant is shown three doors, only one of which hides a valuable prize The contestant chooses one of the doors and the host then opens one of the remaining doors to show that it is empty The host then asks the contestant if she wishes to change her choice of doors from the one she selected to the remaining door Let W denote a door with the prize and E1 and E2 the empty doors Supposing that the contestant switches choices of doors (which, as we will see in a later chapter, she should do), and we write the contestant s initial choice and then the door she nally ends up with, the sample space is S = {(W, E1 ), (W, E2 ), (E1 , W), (E2 , W)}

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EXAMPLE 18

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A Birthday Problem

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A class in calculus has 10 students We are interested in whether or not at least two of the students share the same birthday Here the sample space, showing all possible birthdays, might consist of components with 10 items each We can only show part of the sample space since it contains 36510 = 4196 9 1025 points! Here S = {(March 10, June 15, April 24, ), (May 5, August 2, September 9, )} It may seem counterintuitive, but we can calculate the probability that at least two of the students share the same birthday without enumerating all the points in S We will return to this problem later

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Now we continue to develop the theory of probability

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SOME PROPERTIES OF PROBABILITIES

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Any subset of a sample space is called an event In Example 11, the occurrence of a good item is an event In Example 12, the sample point where the number 3 is to the left of the number 2 is an event In Example 13, the sample point where the rst defective item occurs in an even number of items is an event In Example 14, the sample point where exactly four heads occur is an event We wish to calculate the relative likelihood, or probability, of these events If we try an experiment n times and an event occurs t times, then the relative likelihood of the event is t/n We see that relative likelihoods, or probabilities, are numbers between 0 and 1 If A is an event in a sample space, we write P(A) to denote the probability of the event A Probabilities are governed by these three axioms: 1 P(S) = 1 2 0 P(A) 1 3 If events A and B are P(A B) = P(A) + P(B)

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disjoint,

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that

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A B = ,

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then

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Some Properties of Probabilities

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Axioms 1 and 2 are fairly obvious; the probability assigned to the entire sample space must be 1 since by de nition of the sample space some point in the sample space must occur and the probability of an event must be between 0 and 1 Now if an event A occurs with probability P(A) and an event B occurs with probability P(B) and if the events cannot occur together, then the relative frequency with which one or the other occurs is P(A) + P(B) For example, if a prospective student decides to attend University A with probability 2/5 and to attend University B with probability 1/5, she will attend one or the other (but not both) with probability 2/5 + 1/5 = 3/5 This explains Axiom 3 It is also very useful to consider an event, say A, as being composed of distinct points, say ai ,with probabilities p(ai ) By Axiom 3 we can add these individual probabilities to compute P(A) so

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P(A) =

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p(ai )

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It is perhaps easiest to consider a nite sample space, but our conclusions also apply to a countably in nite sample space Example 14 involved a countable in nite sample space; we will encounter several more examples of these sample spaces in 7 Disjoint events are also called mutually exclusive events Let A denote the points in the sample space where event A does not occur Note that A and A are mutually exclusive so P(S) = P(A A) = P(A) + P(A) = 1 and so we have Fact 1 P(A) = 1 P(A) Axiom 3 concerns events that are mutually exclusive What if they are not mutually exclusive Refer to Figure 11

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