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UT Dallas CS 6313 - Chapter_2_2-4

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Slide 1Sigma-algebraprobabilityProbability: key takeawaysApplication: reliabilitycombinatoricspermutationspermutationscombinationscombinationscombinationsConditional probabilityConditional probabilityBayes Rulelaw of total probabilityBayes RulenextPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapters 2: Probability1SIGMA-ALGEBRAWe will give a rigorous definition of probability and then look at some propertiesFirst, a definition: A sigma-algebra (denoted ) on a sample space is a collection of events such that(a) (the entire sample space is included in the collection)(b) (if an event is included, then so is its complement)(c) (if a finite or countable collection of events is included, then so is its union)Note that a sigma-algebra is just a collection of subsets with these three unique properties•52PROBABILITYAssume a sample space with a sigma-algebra defined on it. Probability is a function such that:(a) (this is possible, since )(b) For any collection of mutually exclusive events , We can immediately conclude that There are a number of properties that can be deduced from this definition … we’ll look at pages 15-17 to see these•53PROBABILITY: KEY TAKEAWAYS and : There will always be some outcome of the experiment.If an event is composed of mutually exclusive outcomes, then the probability of the event is the sum of the probabilities of the outcomes that make it upFor any event E, we have . In other words, the outcome will either be the event or its complement (heads or tails)Events are independent if the probability of their intersection is the product of their probabilitiesBe careful when computing the probability of the union of events – it is the sum of the probabilities only if the events are mutually exclusive•54APPLICATION: RELIABILITYLook at the two examples on pages 18-19 …Why can’t we compute the probability in Example 2.18 the same way that we did in Example 2.19?The devices are used in parallel – we don’t need all three to work, just oneWhy not use the union rule with Example 2.19 – the answer would be .95?We know the events are independent, but not necessarily mutually exclusive – there may be multiple failures. This type of analysis (sequential versus parallel failures) can be used to analyze systems – see Example 2.205COMBINATORICSFor discrete sets of objects, we create events by sampling (i.e. selecting objects)Sampling with replacement means that every sampled object is replaced into the initial set; if there are n objects, the probability of selecting any one of them is 1/n.Sampling without replacement means every sampled item is excluded from further sampling – note this impacts the probabilities of the next selectionObjects are distinguishable if the order they are sampled in yields a different outcome or event; they are indistinguishable if the order is not important. Note this applies to the sample order, not the objects themselves6PERMUTATIONSPossible selections of k distinguishable objects is called a permutation.Note: Order matters in permutationsIf there are n possible objects(objects in the initial set), the number of possible selections of size k from this set with replacement is given byThis is because there are n possibilities for each selection, and there are k selectionsFrom a set consisting of upper and lower case letters and 10 digits, there are trillion passwords for size 8 that can be constructed•57PERMUTATIONSIf there are n possible objects(objects in the initial set), the number of possible selections of size k from this set without replacement is given byThis is because with each selection, we reduce the “pool” by one objectNote that if , we have the classic formula for number of permutations: •58COMBINATIONSWhen order is no longer important, the number of combinations uses a different formula …If we are choosing k objects from a set of n without replacement and the order is not important, the number of ways we can do this isThis is sometime called “n choose k”It is the same formula as the permutation without replacement, but we divide by k! (which is the equivalent of factoring out by the number of orderings)•59COMBINATIONSAs shown in the book, the part of the denominator “takes out” the first k terms of the n! in the numerator, leaving just the last k – 1 terms of n! divided by k!Also helpful to note that …Example: How many poker hands can be drawn from a deck of cards?Answer: Probability of drawing royal flush: Notice this is a combination problem, since the order of the cards in the hand does not matter, and this is without replacement …•510COMBINATIONSThe book shows another example (2.30) on page 25 …Finally, we have the formula for the number of combinations possible when replacement is allowedThis formula is given byThe derivation is on page 26.•511CONDITIONAL PROBABILITYThe conditional probability of an event A given event B is the probability that A occurs when B is known to occur.The notation is .If we think in terms of sets, we know that It then makes sense to define the conditional probability as on the next slide …•512CONDITIONAL PROBABILITY.This is known as the formula for conditional probabilityWe may also write,which gives the general formula for the probability of an intersection (recall the original formula was for independent events)In fact, we now see that A and B are independent if , i.e. the occurrence of B does not affect the probability of A occurring •513BAYES RULEFinding may not be easy Notice that …If we use the formula on the previous slide, we can compute(Bayes Rule) For an example of this rule, see Example 2.33 on page 29In general, , but they are related by Bayes Rule•514LAW OF TOTAL PROBABILITYThe Law of Total Probability can be used to relate the probability of an event to its conditional probabilitiesAs shown on page 30, it assumes that the sample space is partitioned into mutually exclusive and exhaustive events We can compute the probability of A given these events , and since the sample space is composed of the union of these events, we can find the unconditional probability of A this way. Note the formula in 2.10 makes perfectly good sense, and leads to another formulation of Bayes Rule that does not assume knowledge of .Example 2.34 is a classic application of Bayes Rule•515BAYES RULEExample – do problem 2.18. This follows from the second formulation of Bayes Rule given on page 31. (Answer


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UT Dallas CS 6313 - Chapter_2_2-4

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