Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51


 Mervyn Russell
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1 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log ad short. Tables ad give the proportios of shirts sold fallig i the various category combiatios. Table. Relative proportios of shortsleeved shirts sold atter ize laid rit tripe mall Medium arge Table. Relative proportios of logsleeved shirts sold atter ize laid rit tripe mall Medium arge efore we begi, here are some useful terms ad equatios that will help us to address this homewor problem beautifully. We wo t actually use all of these, but they re ice to have for future referece! TERM complemet  The complemet of a evet is the set of all outcomes i the sample space that are ot cotaied i. Complemetary evets are mutually exclusive, ad the sum of their probabilities is always equal to. p. 46 mutually exclusive  Whe evets ad have o outcomes i commo, they are said to be mutually exclusive or disjoit evets. I other words, it is impossible for both ad to occur at the same time. p. 47 exhaustive  Evets,, are exhaustive if oe i must occur, so that, where is the sample space. set of exhaustive evets completely defies its ow sample space p. 73
2 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5 EQUTION The Defiitio of Coditioal robability For ay two evets ad with > 0, the coditioal probability of give that has occurred is defied p. 69 as The Multiplicatio Rule The defiitio of coditioal probability leads us directly to The multiplicatio rule is a algebraic maipulatio of the defiitio of coditioal probability. It may be used to calculate the probability of several stages of idividual evets occurrig i sequece. p. 70 Total law of probability If,, are mutually exclusive ad exhaustive evets, the for ay other evet, The total law of probability may be used to calculate the probability of a particular evet occurrig withi a sequece of a umber of evets. p.73 ayes Theorem If,, is a collectio of mutually exclusive ad exhaustive evets each with i > 0, the for ay other evet for which > 0,,, ayes Theorem is a combiatio of both the defiitio of coditioal probability ad the total law of probability. It allows us to calculate coditioal probabilities for which data provided are ot immediately applicable. p.74
3 Egieerig 33 eautiful Homewor et 3 3 of 7 Kuszmar roblem.5 GORY OF YMO FOR EXREION IN ROIITY Whe cofroted with mathematical expressios i the upcomig solutios, simply substitute the followig words for the symbols idicated, ad everythig will be just fie! is to be replaced with the probability that evet occurs is to be replaced with is equal to is to be replaced with ad is to be replaced with or OK, ow we are ready to defie the evets i the sample space, or i this case, the shirts sold by the departmet store. Each cell i Tables ad represets a mutually exclusive evet. evet may be defied as the sale of a particular type of shirt. The umerical value i the each cell correspods to the probability that the ext shirt sold is of the shirt type associated with that cell. Differet shirt types are distiguished by the choice of sleeve legth, shirt size, ad patter. Note that if we were to add all the values listed i Tables ad, we would obtai a sum of. We may therefore assume that the shirt types represeted defie a exhaustive set. Cosider the followig defiitios: leeve legth {, } {shortsleeved, logsleeved} hirt size {,, 3 } {small, medium, large} atter {,, 3 } {plaid, prit, stripe} Mathematically, the evet that a particular shirt is sold may be see as the itersectio of a particular sleeve legth, shirt size, ad patter. For istace, {shortsleeved, medium, striped shirt} { 3 } cot.
4 Egieerig 33 eautiful Homewor et 3 4 of 7 Kuszmar roblem.5 The probability of a shirt of that type beig sold is listed i Table, ad may be writte as shortsleeved, medium, stripe 3 0. Further ispectio of Tables ad provides us with a umber of useful pieces of iformatio. First, the total umber of shirt types sold by the departmet store must be equal to the total umber of cells for which umerical values are preseted 8. ecod, the tables may be used to calculate the probability of the sale of a shirt for which oly oe or two particular characteristics have bee specified rather tha 3. To do this, we eed oly to add the umerical values i the cells that correspod to the characteristics metioed. For istace, to calculate the probability that the ext shirt sold is a large, we would add all the values i the bottom rows of Tables ad To fid the probability that the ext shirt sold is a shortsleeved plaid, we would add the values i the first colum of Table The last two examples illustrate the use of the total law of probability! Now that all facets of the problem have bee ivestigated ad all ecessary defiitios made, we may retur to the problem statemet ad begi geeratig solutios. This should be a piece of cae! Remember to eep i mid that the shirt types described i this problem represet a set of mutually exclusive ad exhaustive evets. We ca therefore use the total law of probability ad aye s Theorem at will!
5 Egieerig 33 eautiful Homewor et 3 5 of 7 Kuszmar roblem.5 a. What is the probability that the ext shirt sold is a medium, logsleeved, prit shirt? The cell i the secod row ad secod colum of Table cotais the probability we ve bee ased to fid. No problem! The solutio may be writte as 0.5 b. What is the probability that the ext shirt sold is a medium prit shirt? This evet does ot specify oe of the shirt s characteristics sleeve legth so we must tae ito accout the possibility that either a log or shortsleeved shirt may be sold. We will use the total law of probability. ddig the values i the secod rows ad secod colums of Tables ad, we are able to calculate the desired probability. The solutio may be writte as To covert this ad ay other similar mathematical expressios ito a laguage that you ca uderstad, remember to use the woderful Glossary of ymbols for Expressios i robability located o page 3. It really wors! c. What is the probability that the ext shirt sold is a shortsleeved shirt? logsleeved shirt? This questio may be solved i the same maer as part b. This time, either size or patter has bee specified. The probability that the shirt is shortsleeved is equal to the sum of all the values listed i the cells of Table iewise, the sum of all the values i Table represets the probability that the shirt is logsleeved. cot.
6 Egieerig 33 eautiful Homewor et 3 6 of 7 Kuszmar roblem Note that, because there are oly two possible sleeve legths, the evet that the shirt is shortsleeved is the complemet of the evet that the shirt is logsleeved. Therefore, a alterative method for calculatig the probability that the shirt is logsleeved is to subtract the probability that is it shortsleeved from. This method souds much more complicated tha it actually is. Try this d. What is the probability that the size of the ext shirt sold is medium? That the patter of the ext shirt sold is a prit? gai, the total law of probability may be used. However, because the data tables have bee so coveietly arraged, we ca obtai the probability that the ext shirt sold is a medium by fidig the sum of the values i the secod rows of Tables ad The probability that the ext shirt sold is a prit may be foud by addig the secod colums of Tables ad e. Give that the shirt just sold was a short sleeved plaid, what is the probability that its size was medium? The desired probability is a coditioal probability. The defiitio of coditioal probability gives us cot.
7 Egieerig 33 eautiful Homewor et 3 7 of 7 Kuszmar roblem.5 Expadig the deomiator usig the total law of probability, the expressio becomes 3 Fially, icorporatig the appropriate values from Table, we obtai f. Give that the shirt just sold was a medium plaid, what is the probability that it was short sleeved? og sleeved? I the same maer as part e, the defiitio of coditioal probability may be used i cojuctio with the total law of probability ad the data available i Tables ad. The probability that the shirt was shortsleeved, give that it was a medium plaid is The probability that the shirt was logsleeved is Oce agai, the fact that is the complemet of provides us with a alterative method for calculatig the probability above
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