Using Counting Techniques to Determine Probabilities

Transcription

1 Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe of possible outcomes whe multiple evets ae cosideed. I this lesso, we ae tyig to develop coutig techiques we ca use to help compute pobabilities. Oe of the most fudametal coutig techiques is the fudametal coutig piciple (sometimes efeed to as the multiplicatio axiom). efoe we itoduce the fudametal coutig piciple, let s look at some examples. Example If a woma has two blouses {, } ad thee skits {s, s, s }, how may diffeet outfits cosistig of a blouse ad a skit ca she wea? We ca use a tee diagam to detemie the umbe of diffeet outfits. s s s s s s s s s s s s Notice we ed up with two sets (o goups) of thee outfits (thee with the fist blouse ad thee with the secod blouse). We ca also use the multiplicatio fact to detemie the umbe of outfits because the meaig of is two goups of thee. The pocedue we just employed is called the fudametal coutig piciple. The fudametal coutig piciple states that if oe task ca be doe i m ways, ad a secod task ca be doe i ways, the the opeatio ivolvig the fist task

2 followed by the secod ca be pefomed i m ways. The fudametal coutig piciple is ot limited to just two tasks; it ca be used fo ay umbe of tasks. Example tuck licese plate cosists of a lette followed by fou digits. How may such licese plates ae possible? Sice thee ae 6 possibilities fo the lette ad 0 possibilities fo each of the fou digits, the fudametal coutig piciple gives us = 60,000 possible licese plates. Detemiig obability with the Fudametal outig iciple Example tuck licese plate cosists of a lette followed by fou digits. If you select oe licese plate at adom, what is the pobability it cotais the lette Z? Recall that the defiitio of pobability is. I example, we detemied that thee ae 60,000 possible outcomes. Now, the umbe of acceptable outcomes is detemied by the fudametal coutig piciple as = 0,000. (Note thee is oly oe choice fo the lette, that beig Z.) 0,000 So, (licese plate cotais Z) =. 60,000 6

3 emutatios Suppose we ae asked to fid the sequeces that ca be fomed with the lettes {,, } if o lette is epeated. Fist let s use a tee diagam to detemie the umbe of sequeces: The tee diagam gives us six aagemets. We ca also use the fudametal coutig piciple to detemie the umbe of sequeces. Sice the lettes caot be epeated, we have thee choices fo the fist lette, two choices fo the secod lette, ad oe choice fo the thid lette. Theefoe, the fudametal coutig piciple gives us = 6. We ca wite this equatio usig factoial otatio as! = 6. Factoial otatio is defied as! = ( ) ( ). Two impotat defiitios ae! = ad 0! =. agemets like these, whee ode is impotat ad o elemet is epeated, ae called pemutatios. pemutatio of a set of elemets is a odeed aagemet whee each elemet is used oce. Example Give five lettes {,,, D, E}, fid the umbe of fou-lette sequeces possible. Fom the fudametal coutig piciple, we have 5 =.

4 We ofte ecoute situatios like this oe, whee we have a set of objects ad we ae selectig objects to fom pemutatios. We efe to this as pemutatios of objects take at a time, ad we wite it as. Theefoe, example ca also be asweed as: The umbe of fou-lette sequeces is 5 =. Let s defie. The umbe of pemutatios of objects take at a time is: ) ) )... ) (which is the same as the fudametal coutig piciple), o!, whee ad ae atual umbes. )! We ca show that this fomula is the same as the fist oe:! )! ) )... ) ) )... ) ) ) )... ). You should udestad both fomulas ad feel comfotable applyig eithe oe. Example 5 ompute 6 usig both fomulas. Fist fomula: 6 5 =. 6! (6 5 ) Secod fomula: 6 5.! ( ) Example 6 You have math books ad 5 histoy books to put o a shelf that has 5 slots. I how may ways ca the books be shelved if the fist thee slots ae filled with math books ad the ext two slots ae filled with histoy books? Fist we detemie the umbe of ways the math books ca be aaged. This is a pemutatio of objects take at a time, o : = =. Next we detemie the umbe of ways the histoy books ca be aaged. This is a pemutatio of 5 objects take at a time, o 5 : 5 = 5 =.

5 Now we use the fudametal coutig piciple. Sice thee ae ways to aage the math books ad ways to aage the histoy books, the total umbe of ways to aage the math ad histoy books is, o 80. ombiatios Suppose that we have a set of thee lettes {,, }, ad we ae asked to make twolette sequeces. We have six pemutatios: Now suppose we have a goup of thee people {,, }, l, ob, ad his, espectively, ad we ae asked to fom committees of two people each. This time we have oly thee committees: Whe fomig committees, the ode is ot impotat because the committee that has l ad ob is o diffeet fom the committee that has ob ad l. s a esult, we have oly thee committees, ot six. Fomig sequeces is a example of pemutatios, while fomig committees is a example of combiatios. emutatios ae those aagemets whee ode is impotat, while combiatios ae those aagemets whee ode is ot sigificat. Recall that epesets the umbe of combiatios of objects take at a time. Now let s look at a situatio i which we eed to detemie the elatioship betwee the umbe of combiatios ad the umbe of pemutatios. Suppose we wat to detemie the umbe of pemutatios of lettes {,,, D} take at a time ( ). We kow that =, ad list all pemutatios: D D D D D D D D D D D D D D D D D D 5

6 If we wat to detemie the umbe of combiatios of lettes {,,, D} take at a time, we see that thee ae oly fou sice ode does ot matte. To detemie the umbe of combiatio of objects take at a time, we ca fist detemie the umbe of pemutatios of objects take at a time ad the divide by the umbe of pemutatios fo ay objects take at a time. So i this case, 6. We divide by 6 ( ) because oce we have the thee specific lettes, we ae ot coceed with the ode; so, we divide out the 6 diffeet odes of lettes take at a time. The fomal defiitio of a combiatio is a set of elemets i a aagemet whee each elemet is used oce ad ode is ot impotat. The umbe of combiatios of objects take at a time is witte..!, whee ad ae atual umbes. )!.! Notice that Example 7 ompute 5.! gives )! ad! gives. 5 (5 5! )! 5!!! 5 ( )( ) 0. ombiatios Ivolvig Seveal Sets So fa we have solved the basic combiatio poblem of objects chose fom diffeet objects. Now we will coside moe-complex poblems. Example 8 How may five-peso committees cosistig of me ad wome ca be chose fom a goup of me ad wome? Fist we coside the umbe of combiatios of me chose at a time ( ): = 6. Next we coside the umbe of combiatios of wome chose at a time ( ): =. 6

7 Sice thee ae 6 ways of choosig me ad ways of choosig wome, the fudametal coutig piciple gives us the umbe of five-peso committees: 6 =. obabilities Ivolvig ombiatios Example 9 high school club cosists of feshme, 5 sophomoes, 5 juios, ad 6 seios. What is the pobability that a committee of people chose at adom icludes oe studet fom each class? The total umbe of committees possible is sice thee ae studets i all ad we wat to select : =,85. The umbe of combiatios of feshma take at a time is =. The umbe of combiatios of 5 sophomoes take at a time is 5 = 5. The umbe of combiatios of 5 juios take at a time is 5 = 5. The umbe of combiatios of 6 juios take at a time is 6 = 6. Usig the fudametal coutig piciple, the umbe of committees is = 600. Example 0 high school club cosists of feshme, 5 sophomoes, 5 juios, ad 6 seios. What is the pobability that a committee of people chose at adom icludes at least oe seio? The total umbe of committees possible is sice thee ae studets i all ad we wat to select : =,85. We wat the committee to cotai at least oe seio, so it could cotai exactly oe seio, exactly two seios, exactly thee seios, o exactly fou seios. Fidig the pobability that the committee cotais at least oe seio equies us to fid the pobability fo each of the fou cases ad the use the additio piciple sice these fou possibilities ae disjoit evets. 6 6,85 8,85 (exactly oe seio) = (exactly two seios) = ,85 65,85 7

8 ,85 80,85 (exactly thee seios) = (exactly fou seios) = ,85 5,85 We kow that (at least oe seio) = (exactly oe seio) + (exactly two seios) + (exactly thee seios) + (exactly fou seios), so: (at least oe seio) = = We ca compute this pobability i a much moe efficiet mae, howeve, by usig the complemet of the evet at least oe seio. The complemet is that the committee cotais o seios (the oly othe possible outcome): (at least oe seio) = (o seios) (at least oe seio) 6 0,00, Whe dealig with pobability situatios ivolvig at least oe, the most efficiet way to detemie the pobability is to use the complemetay evet oe. This kowledge aticle is adapted fom the followig souce: Sekho, Rupide. "Sets ad outig." oexios. July,. 8