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Question Number 149440 by nadovic last updated on 05/Aug/21

Answered by Olaf_Thorendsen last updated on 07/Aug/21

(a) A_1 = “persons from only one of the three groups”. That means 3 volleyball players or 3 hockey players. P(A_1 ) = (C_3 ^3 /C_3 ^9 )+(C_3 ^4 /C_3 ^9 ) = (5/(84)) ≈ 5,95%. (b) A_2 = “two persons from one group and one person from another group”. That means −1 football player and 2 volleyball players or −2 football players and 1 volleyball player or −1 football player and 2 hockey players or −2 football players and 1 hockey player or −1 volleyball player and 1 hockey players or −2 volleyball players and 1 hockey player. P(A_2 ) = ((C_1 ^2 C_2 ^3 )/C_3 ^9 )+((C_2 ^2 C_1 ^3 )/C_3 ^9 )+ ((C_1 ^2 C_2 ^4 )/C_3 ^9 )+ ((C_2 ^2 C_1 ^4 )/C_3 ^9 ) + ((C_1 ^3 C_2 ^4 )/C_3 ^9 )+((C_2 ^3 C_1 ^4 )/C_3 ^9 ) P(A_2 ) = (6/(84))+(3/(84))+ ((12)/(84))+ (4/(84))+((18)/(84))+((12)/(84)) P(A_2 ) = ((55)/(84)) ≈ 65,48%.

$$\left(\mathrm{a}\right)\:\mathrm{A}_{\mathrm{1}} =\:``\mathrm{persons}\:\mathrm{from}\:\mathrm{only}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three} \\ $$$$\mathrm{groups}''. \\ $$$$\mathrm{That}\:\mathrm{means}\:\mathrm{3}\:\mathrm{volleyball}\:\mathrm{players}\:\mathrm{or} \\ $$$$\mathrm{3}\:\mathrm{hockey}\:\mathrm{players}. \\ $$$$\mathrm{P}\left(\mathrm{A}_{\mathrm{1}} \right)\:=\:\frac{\mathrm{C}_{\mathrm{3}} ^{\mathrm{3}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }+\frac{\mathrm{C}_{\mathrm{3}} ^{\mathrm{4}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }\:=\:\frac{\mathrm{5}}{\mathrm{84}}\:\approx\:\mathrm{5},\mathrm{95\%}. \\ $$$$ \\ $$$$\left(\mathrm{b}\right)\:\mathrm{A}_{\mathrm{2}} \:=\:``\mathrm{two}\:\mathrm{persons}\:\mathrm{from}\:\mathrm{one}\:\mathrm{group} \\ $$$$\mathrm{and}\:\mathrm{one}\:\mathrm{person}\:\mathrm{from}\:\mathrm{another}\:\mathrm{group}''. \\ $$$$\mathrm{That}\:\mathrm{means} \\ $$$$−\mathrm{1}\:\mathrm{football}\:\mathrm{player}\:\mathrm{and}\:\mathrm{2}\:\mathrm{volleyball}\:\mathrm{players}\:\mathrm{or} \\ $$$$−\mathrm{2}\:\mathrm{football}\:\mathrm{players}\:\mathrm{and}\:\mathrm{1}\:\mathrm{volleyball}\:\mathrm{player}\:\mathrm{or} \\ $$$$−\mathrm{1}\:\mathrm{football}\:\mathrm{player}\:\mathrm{and}\:\mathrm{2}\:\mathrm{hockey}\:\mathrm{players}\:\mathrm{or} \\ $$$$−\mathrm{2}\:\mathrm{football}\:\mathrm{players}\:\mathrm{and}\:\mathrm{1}\:\mathrm{hockey}\:\mathrm{player}\:\mathrm{or} \\ $$$$−\mathrm{1}\:\mathrm{volleyball}\:\mathrm{player}\:\mathrm{and}\:\mathrm{1}\:\mathrm{hockey}\:\mathrm{players}\:\mathrm{or} \\ $$$$−\mathrm{2}\:\mathrm{volleyball}\:\mathrm{players}\:\mathrm{and}\:\mathrm{1}\:\mathrm{hockey}\:\mathrm{player}. \\ $$$$\mathrm{P}\left(\mathrm{A}_{\mathrm{2}} \right)\:=\:\frac{\mathrm{C}_{\mathrm{1}} ^{\mathrm{2}} \mathrm{C}_{\mathrm{2}} ^{\mathrm{3}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }+\frac{\mathrm{C}_{\mathrm{2}} ^{\mathrm{2}} \mathrm{C}_{\mathrm{1}} ^{\mathrm{3}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }+\:\frac{\mathrm{C}_{\mathrm{1}} ^{\mathrm{2}} \mathrm{C}_{\mathrm{2}} ^{\mathrm{4}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }+\:\frac{\mathrm{C}_{\mathrm{2}} ^{\mathrm{2}} \mathrm{C}_{\mathrm{1}} ^{\mathrm{4}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} } \\ $$$$+\:\frac{\mathrm{C}_{\mathrm{1}} ^{\mathrm{3}} \mathrm{C}_{\mathrm{2}} ^{\mathrm{4}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} }+\frac{\mathrm{C}_{\mathrm{2}} ^{\mathrm{3}} \mathrm{C}_{\mathrm{1}} ^{\mathrm{4}} }{\mathrm{C}_{\mathrm{3}} ^{\mathrm{9}} } \\ $$$$\mathrm{P}\left(\mathrm{A}_{\mathrm{2}} \right)\:=\:\frac{\mathrm{6}}{\mathrm{84}}+\frac{\mathrm{3}}{\mathrm{84}}+\:\frac{\mathrm{12}}{\mathrm{84}}+\:\frac{\mathrm{4}}{\mathrm{84}}+\frac{\mathrm{18}}{\mathrm{84}}+\frac{\mathrm{12}}{\mathrm{84}} \\ $$$$\mathrm{P}\left(\mathrm{A}_{\mathrm{2}} \right)\:=\:\frac{\mathrm{55}}{\mathrm{84}}\:\approx\:\mathrm{65},\mathrm{48\%}. \\ $$

Commented by nadovic last updated on 07/Aug/21

Thank you Sir

$${Thank}\:{you}\:{Sir} \\ $$

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