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  • A, B and C are contesting the election for the post of secretary of a club which does not allow ladies to become members. The probabilities of A,B and C winning the election are

A, B and C are contesting the election for the post of secretary of a club which does not allow ladies to become members. The probabilities of A,B and C winning the election are \frac{1}{3}, \frac{2}{9}\text{ and } \frac{4}{9} respectively. The probabilities of introducing the clause of admitting lady members to the club by A, B, and C are 0.6,0.7 and 0.5 respectively. The probability that ladies will be taken as members in the club after the election is

Option: 1

\frac{26}{45}


Option: 2

\frac{5}{9}


Option: 3

\frac{19}{45}


Option: 4

None of these


Answers (1)

best_answer

Let \mathrm{E}_{\mathrm{A}}= the event of A becoming secretary.

Similarly, \mathrm{E}_{\mathrm{B}} \text{ and } \mathrm{E}_{\mathrm{C}}.

\mathrm{E_L=} the event of admitting lady members.

Here, \mathrm{P\left(E_{\mathrm{A}}\right)=\frac{1}{3}, P\left(E_{\mathrm{B}}\right)=\frac{2}{9}, P\left(E_{\mathrm{C}}\right)=\frac{4}{9}}

Clearly, \mathrm{ E_A, E_B, E_C }  are mutually exclusive and exhaustive. 

Also,\mathrm{P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{A}}}\right)=0.6, \quad P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{B}}}\right)=0.7, \quad P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{C}}}\right)=0.5 }

\therefore the required probability

\mathrm{=P\left(E_{\mathrm{A}}\right) \cdot P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{A}}}\right)+P\left(E_{\mathrm{B}}\right) \cdot P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{B}}}\right)+P\left(E_{\mathrm{C}}\right) \cdot P\left(\frac{E_{\mathrm{L}}}{E_{\mathrm{C}}}\right)}

=\frac{1}{3} \times \frac{3}{5}+\frac{2}{9} \times \frac{7}{10}+\frac{4}{9} \times \frac{5}{10}=\frac{26}{45}

 

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Anam Khan

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