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Question Number 173842 by byaw last updated on 19/Jul/22
  The probability that an athlete will not win any of the three races is 1/4. 
  If the athlete runs in all the races, what is the probability that the athlete will win: 
  (I) only the second race:  
  (ii) all the three races   
  (iii) only two of the races?
$$ \\ $$The probability that an athlete will not win any of the three races is 1/4.
If the athlete runs in all the races, what is the probability that the athlete will win:
(I) only the second race:

(ii) all the three races

(iii) only two of the races?

Answered by mr W last updated on 20/Jul/22
say the probability that he wins a  race is p. it is given  (1−p)^3 =(1/4)   ⇒p=1−(1/( (4)^(1/3) ))=0.37  (i)  0.63×0.37×0.63=0.146=14.6%  (ii)  0.37^3 =0.05=5%  (iii)  3×0.37^2 ×0.63=0.259=25.9%
$${say}\:{the}\:{probability}\:{that}\:{he}\:{wins}\:{a} \\ $$$${race}\:{is}\:{p}.\:{it}\:{is}\:{given} \\ $$$$\left(\mathrm{1}−{p}\right)^{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{4}}\: \\ $$$$\Rightarrow{p}=\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{4}}}=\mathrm{0}.\mathrm{37} \\ $$$$\left({i}\right) \\ $$$$\mathrm{0}.\mathrm{63}×\mathrm{0}.\mathrm{37}×\mathrm{0}.\mathrm{63}=\mathrm{0}.\mathrm{146}=\mathrm{14}.\mathrm{6\%} \\ $$$$\left({ii}\right) \\ $$$$\mathrm{0}.\mathrm{37}^{\mathrm{3}} =\mathrm{0}.\mathrm{05}=\mathrm{5\%} \\ $$$$\left({iii}\right) \\ $$$$\mathrm{3}×\mathrm{0}.\mathrm{37}^{\mathrm{2}} ×\mathrm{0}.\mathrm{63}=\mathrm{0}.\mathrm{259}=\mathrm{25}.\mathrm{9\%} \\ $$

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