ENGI 4421 Counting Techniques for Probability Page Example 3.01 [Navidi Section 2.2; Devore Section 2.3]

Transcription

1 ENGI 4421 Coutig Techiques fo Pobability Page 3-01 Example 3.01 [Navidi Sectio 2.2; Devoe Sectio 2.3] Fou cads, labelled A, B, C ad D, ae i a u. I how may ways ca thee cads be daw (a) with eplacemet? (b) without eplacemet? (c) without eplacemet (if the ode of selectio does t matte)? (a) With eplacemet meas that, each time a cad is daw, it is put back ito the u befoe the ext cad is daw. Theefoe the same cad ca be withdaw moe tha oce fom the u. The complete sample space is listed hee. AAA AAB AAC AAD ABA ABB ABC ABD ACA ACB ACC ACD ADA ADB ADC ADD BAA BAB BAC BAD BBA BBB BBC BBD BCA BCB BCC BCD BDA BDB BDC BDD CAA CAB CAC CAD CBA CBB CBC CBD CCA CCB CCC CCD CDA CDB CDC CDD DAA DAB DAC DAD DBA DBB DBC DBD DCA DCB DCC DCD DDA DDB DDC DDD I geeal, # ways to choose objects fom with eplacemet is

2 ENGI 4421 Coutig Techiques fo Pobability Page 3-02 (b) Without eplacemet meas that each time a cad is daw, it stays out of the u fo all subsequet dawigs of cads. The same cad caot be daw moe tha oce. Idetify the educed sample space fom pat (a): AAA AAB AAC AAD ABA ABB ABC ABD ACA ACB ACC ACD ADA ADB ADC ADD BAA BAB BAC BAD BBA BBB BBC BBD BCA BCB BCC BCD BDA BDB BDC BDD CAA CAB CAC CAD CBA CBB CBC CBD CCA CCB CCC CCD CDA CDB CDC CDD DAA DAB DAC DAD DBA DBB DBC DBD DCA DCB DCC DCD DDA DDB DDC DDD I geeal, the umbe of ways i which objects ca be daw fom objects without eplacemet (with the ode of selectio beig impotat) is the umbe of pemutatios: P Defiitio the factoial fuctio, fo all positive iteges, is!

3 ENGI 4421 Coutig Techiques fo Pobability Page 3-03 Theefoe the umbe of pemutatios of objects fom objects is also P Alteative symbols fo pemutatios iclude P, P (c) If oly the idetities of the cads daw fom the u mattes, ot the ode i which they wee daw, the we ae seekig the umbe of combiatios of 3 cads fom 4. The pemutatios ABC, ACB, BAC, ae all the same combiatio, because all of these pemutatios cotai the same thee cads { A, B, C }. Howeve the pemutatio ABD is a diffeet combiatio fom the pemutatio BAC, because oe cad is diffeet (D istead of C). The sigle combiatio { A, B, C } cotais a umbe of pemutatios equal to the umbe of ways i which the thee lettes ca be e-aaged amog themselves: Moe geeally, a sigle combiatio of objects fom objects cotais pemutatios. Theefoe the umbe of combiatios of objects fom objects (the umbe of ways of dawig them without eplacemet ad with the ode of selectio beig ielevat) is Note that this defiitio is cosistet oly if 0! = 1.

4 ENGI 4421 Coutig Techiques fo Pobability Page 3-04 Example 3.02 Fou cads, labelled A, B, C ad D, ae i a u. I how may ways ca two cads be daw (a) with eplacemet? (b) without eplacemet? (c) without eplacemet (if the ode of selectio does t matte)? Fo pat (a) of this questio, the complete sample space is listed below. AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD (a) (b) (c) The combiatios ae AB AC AD BC BD CD The pemutatios ae AB BA AC CA AD DA BC CB BD DB CD DC

5 ENGI 4421 Coutig Techiques fo Pobability Page 3-05 Example 3.03 Evaluate 11 6 Example 3.04 Evaluate P 2, 9 Example 3.05 Simplify Example 3.06 Simplify

6 ENGI 4421 Coutig Techiques fo Pobability Page 3-06 Also ote the idetities 0 P0 P 0, C 0 C 1 P P!, P P!, 1 1, ad P1 P 1, C 1 C Summay: The umbe of ways to daw objects fom distiguishable objects is: [with eplacemet (odeed):] [without eplacemet (odeed):] [without eplacemet (uodeed):] P C The case with eplacemet (uodeed) seldom aises i pactice, but the umbe of ways of dawig objects fom distiguishable objects i this case ca be show to be 1 C. See fo a illustatio of these values fo some choices of ad. Example 3.07 (a) (b) I how may ways ca a team of thee me ad thee wome be chose fom a goup of five me ad six wome? I how may ways ca a team of thee me ad thee wome be chose fom a goup of five me ad six wome whe the team has oe leade, oe othe membe ad a eseve fo the me ad likewise fo the wome? (a)

7 ENGI 4421 Coutig Techiques fo Pobability Page 3-07 Example 3.07 (cotiued) (b) Also ote the idetity which leads to Pascal s tiagle: 1 C C C1 The umbe of combiatios also appeas i the biomial expasio a b is a atual umbe: a b a C a b C a b C a b C a b b o k k k k 0 a b C a b, whee

8 ENGI 4421 Coutig Techiques fo Pobability Page 3-08 Example z x j y Example x 4 [Ed of Chapte 3]