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Question Number 105812 by Ar Brandon last updated on 31/Jul/20

In a by-election 20% of the electorate voted for Mr X.  If 5 voters are chosen at random from this electorate, what  is the probability that 20% of the sample voted for Mr X ?  (a) 1    (b) (1/5)    (c) (1/(625))    (d) ((256)/(625))

$$\mathrm{In}\:\mathrm{a}\:\mathrm{by}-\mathrm{election}\:\mathrm{20\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{electorate}\:\mathrm{voted}\:\mathrm{for}\:\mathrm{Mr}\:\mathrm{X}. \\ $$$$\mathrm{If}\:\mathrm{5}\:\mathrm{voters}\:\mathrm{are}\:\mathrm{chosen}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{this}\:\mathrm{electorate},\:\mathrm{what} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{20\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sample}\:\mathrm{voted}\:\mathrm{for}\:\mathrm{Mr}\:\mathrm{X}\:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{1}\:\:\:\:\left(\mathrm{b}\right)\:\frac{\mathrm{1}}{\mathrm{5}}\:\:\:\:\left(\mathrm{c}\right)\:\frac{\mathrm{1}}{\mathrm{625}}\:\:\:\:\left(\mathrm{d}\right)\:\frac{\mathrm{256}}{\mathrm{625}} \\ $$

Answered by Don08q last updated on 31/Jul/20

a = P(X) = ((20)/(100))                       = (1/5)  b = P(X^− ) = 1 − ((20)/(100))                       = (4/5)  20% of 5 voters = 1   P(X = 1) =^5 C_1 ((1/5))^1 ((4/5))^4                        = ((256)/(625))   The answer is D

$${a}\:=\:\mathrm{P}\left(\mathrm{X}\right)\:=\:\frac{\mathrm{20}}{\mathrm{100}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{5}} \\ $$$${b}\:=\:\mathrm{P}\left(\overset{−} {\mathrm{X}}\right)\:=\:\mathrm{1}\:−\:\frac{\mathrm{20}}{\mathrm{100}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\mathrm{20\%}\:{of}\:\mathrm{5}\:{voters}\:=\:\mathrm{1} \\ $$$$\:\mathrm{P}\left(\mathrm{X}\:=\:\mathrm{1}\right)\:=\:^{\mathrm{5}} {C}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{5}}\right)^{\mathrm{1}} \left(\frac{\mathrm{4}}{\mathrm{5}}\right)^{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{256}}{\mathrm{625}} \\ $$$$\:{The}\:{answer}\:{is}\:\mathrm{D} \\ $$

Commented by Ar Brandon last updated on 31/Jul/20

Thanks

$$\mathrm{Thanks} \\ $$

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