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Probability and Random Variables 145
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example: In the experiment of flipping two coins, let the event A = obtain at least one head The sample space contains four elements ({HH, HT, TH, TT}) s = 3 because there are three ways for our outcome to be considered a success ({HH, HT, TH}) and f = 1 Thus P( A ) = 3 3 = 3+1 4
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example: Consider rolling two fair dice and noting their sum A sample space for this event can be given in table form as follows:
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Let B = the sum of the two dice is greater than 4 There are 36 outcomes in the samples space, 30 of which are greater than 4 Thus, P(B) = Furthermore, 30 5 = 36 6
= P ( 2) + P (3) +
+ P (12) =
1 2 + + 36 36
1 = 1 36
Probabilities of Combined Events
P(A or B): The probability that either event A or event B occurs (They can both occur, but only one needs to occur) Using set notation, P(A or B) can be written P( A B) A B is spoken as, A union B P(A and B): The probability that both event A and event B occur Using set notation, P(A and B) can be written P( A B) A B is spoken as, A intersection B example: Roll two dice and consider the sum (see table) Let A = one die shows a 3, B = the sum is greater than 4 Then P(A or B) is the probability that either one die shows a 3 or the sum is greater than 4 Of the 36 possible outcomes in the sample space, there are 32 possible outcomes that are successes [30 outcomes greater than 4 as well as (1,3) and (3,1)], so P( A or B) = 32 36
146 U Step 4 Review the Knowledge You Need to Score High
There are nine ways in which a sum has one die showing a 3 and has a sum greater than 4: [(3,2), (3,3), (3,4), (3,5), (3,6), (2,3), (4,3), (5,3), (6,3)], so P( A and B) = 9 36
Complement of an event A: events in the sample space that are not in event A The complement of an event A is symbolized by A , or A c Furthermore, P( A ) = 1 P (A)
Mutually Exclusive Events
Mutually exclusive (disjoint) events: Two events are said to be mutually exclusive (some texts refer to mutually exclusive events as disjoint) if and only if they have no outcomes in common That is, A B = If A and B are mutually exclusive, then P(A and B) = P ( A B ) = 0 example: in the two-dice rolling experiment, A = face shows a 1 and B = sum of the two dice is 8 are mutually exclusive because there is no way to get a sum of 8 if one die shows a 1 That is, events A and B cannot both occur
Conditional Probability
Conditional Probability: The probability of A given B assumes we have knowledge of an event B having occurred before we compute the probability of event A This is symbolized by P(A|B) Also, P ( A | B) = P ( A and B) P (B )
Although this formula will work, it s often easier to think of a condition as reducing, in some fashion, the original sample space The following example illustrates this shrinking sample space example: Once again consider the possible sums on the roll of two dice Let A = the sum is 7, B = one die shows a 5 We note, by counting outcomes in the table, that P(A) = 6/36 Now, consider a slightly different question: what is P(A|B) (that is, what is the probability of the sum being 7 given that one die shows a 5) solution: Look again at the table:
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