Slide 1ProbabilityToolsTools: setsTools: setsTools: setsNext …Probability and Statistics in Computer Science and Software Engineering Chapters 2: Probability1ProbabilityIn our normal lives, we understand probabilit y to mean the chance of an event occurring …Flip a (fair) coin – there is a 50-50 chance that it will come up headsMore accurately, if we flip the coin n time and record the outcome of each event, we expect the particular outcom e (heads) to occur approximately ½ of the timeIn other words, for 150 random flips, we should count about 75 heads (and 75 tails)In the long run, we would expect the proportion of heads to be very close to ½ or .500 – in fact, the more flips we add to the experiment, the closer we would expect to be to 50% headsWe instinctively do these computations all the time: Pick a day of the week at random, there is a 1-in-7 chance that you land on a Wednesday …2ToolsA collection of elementary results, or outcomes, is called a sample space.In the previous example (coin flip), the sample space consisted of two outcomes: a heads or a tails. Any set of outcomes is called an event. An event is a subset of the sample spaceExample: Roll a dice. There are 6 possible outcomes (denoted by the numbers on the face of the dice)Events could be roll an even number, roll and odd number, roll a number less than 5, etc …3Tools: setsHow many events are possible? If the sample space has N possible outcomes, then there are a total of possible eventsFor the coin flip example, there are 4 possible events …We know two of them – coin comes up heads, coin comes up tails – what are the other two?One is “some outcome occurs” – this is the equivalent of the entire sample space. Always trueThe other is “no outcome occurs” – this is equivalent to the null subset, which has nothing in it. Always falseFor any event, there are two possibilities for each outcome (part of the event or not) … since there are N outcomes, we end up with possible events•=4Tools: setsCommon notation: = sample space = empty event (no outcomes included) = probability of event E occurring Consider a game between the Cowboys and the Giants … = {Cowboys win, Giants win, they tie}There are actually eight events associated with this sample space …Cowboys win, lose, tie, get at least a tie, get at most a tie, get some result, get no result•=5Tools: setsNotice that there is a connection between probability and set theory …If we interpret the sample space as a set and an event as a subset, then in some sense the “probability” of the event occurring should be related to the proportion of the event subset to the entire sample spaceIn other words, if the event only consists of a small portion of the sample space, we would expect the event to be “rare”But if the event consists of a large portion of the sample space, we would expect the event to be likely …We need to understand set theory … see pages 11-13.6Next …Read chapters 2.2 – 2.4 …Intro to ProbabilityCombinatorics – permutations and combinationsConditional probability and independence – Bayes
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