49207908-slides

# 49207908-slides - Lecture Notes on PROBABILITY and...

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Lecture Notes on PROBABILITY and STATISTICS Eusebius Doedel http://cmvl.cs.concordia.ca/courses/comp-233/winter-2011/

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SAMPLE SPACES DEFINITION : The sample space is the set of all possible outcomes of an experiment. EXAMPLE: When we ﬂip a coin then sample space is S = { H , T } , where H denotes that the coin lands ”Heads up” and T denotes that the coin lands ”Tails up”. For a ” fair coin ” we expect H and T to have the same ” chance ” of occurring, i.e. , if we ﬂip the coin many times then about 50 % of the outcomes will be H . We say that the probability of H to occur is 0.5 (or 50 %) . The probability of T to occur is then also 0.5. 1
EXAMPLE: When we toss a fair die then the sample space is S = { 1 , 2 , 3 , 4 , 5 , 6 } . The probability the die lands with k up is 1 6 , ( k = 1 , 2 , · · · , 6). When we toss it 1200 times we expect a 5 up about 200 times. The probability the die lands with an even number up is 1 6 + 1 6 + 1 6 = 1 2 . 2

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EXAMPLE: When we toss a coin 3 times and record the results in the sequence they occur, then the sample space is S = { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } . Elements of S are ” vectors ”, ” sequences ”, or ” ordered outcomes ”. We may expect each of the 8 outcomes to be equally likely. Thus the probability of the sequence HTT is 1 8 . The probability of a sequence to contain precisely two Heads is 1 8 + 1 8 + 1 8 = 3 8 . 3
EXAMPLE: When we toss a coin 3 times and record the results without paying attention to the order in which they occur, e.g. , if we only record the number of Heads, then the sample space is S = { H, H, H } , { H, H, T } , { H, T, T } , { T, T, T } . The outcomes in S are now sets ; i.e. , order is not important. Recall that the ordered outcomes are { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } . Note that { H, H, H } corresponds to one of the ordered outcomes, { H, H, T } ,, three ,, { H, T, T } ,, three ,, { T, T, T } ,, one ,, Thus { H, H, H } and { T, T, T } each occur with probability 1 8 , while { H, H, T } and { H, T, T } each occur with probability 3 8 . 4

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Events In Probability Theory subsets of the sample space are called events . EXAMPLE: The set of basic outcomes of tossing a die once is S = { 1 , 2 , 3 , 4 , 5 , 6 } , so the subset E = { 2 , 4 , 6 } is an example of an event. If a die is tossed once and it lands with a 2 or a 4 or a 6 up then we say that the event E has occurred . We have already seen that the probability that E occurs is P ( E ) = 1 6 + 1 6 + 1 6 = 1 2 . 5
The Algebra of Events Since events are sets , namely, subsets of the sample space S , we can do the usual set operations : If E and F are events then we can form E c the complement of E E F the union of E and F EF the intersection of E and F We write E F if E is a subset of F . REMARK. In Probability Theory we use E c instead of ¯ E , EF instead of E F . 6

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If the sample space S is finite then we typically allow any subset of S to be an event. EXAMPLE: If we randomly draw one character from a box contain- ing the characters a , b , and c , then the sample space is S = { a , b , c } , and there are 8 possible events, namely, those in the set of events E = { } , { a } , { b } , { c } , { a, b } , { a, c } , { b, c } , { a, b, c } .
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