Question

Given that she does not achieve success, the chance she studied for 4 hour, is

• A. $\frac{18}{26}$
• B. $\frac{19}{26}$
• C. $\frac{20}{26}$
• D. $\frac{21}{26}$

Answer 1

The correct option is D $\frac{21}{26}$

Given,The probability of a Jee Aspirant to be successful if he studies for 10 hours per day,$P\left(t_{10}\right)=0.8$
The probability of a Jee Aspirant to be successful if he studies for 7 hours per day,$P\left(t_7\right)=0.6$
The probability of a Jee Aspirant to be successful if he studies for 4 hours per day,$P\left(t_4\right)=0.4$
The probability that she will study 10 hours per day,$P\left(A\right)=0.1$
The probability that she will study 7 hours per day,$P\left(B\right)=0.2$
The probability that she will study 4 hours per day,$P\left(C\right)=0.7$
$\therefore Theprobabilitythatshewillbesuccessful,P\left(S\right)=P\left(t_{10}\right)\ast P\left(A\right)+P\left(t_7\right)\ast P\left(B\right)+P\left(t_4\right)\ast P\left(S/t_4\right)$
$=0.8\ast 0.1+0.6\ast 0.2+0.4\ast 0.7=0.48$
$\therefore$ The probability that she will not success,$P\left(\overline{S}\right)=1-P\left(S\right)=1-0.48=0.52$
But probability that she studies 4 hours per day but didn't success=$P({\overline{t}}_4\cap C)=P\left({\overline{t}}_4\right)P\left(C\right)=(1-0.4)\ast 0.7=0.42$
Since $(A,t_{10}),(B,t_7),(C,t_4)$ pairwise independent $\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)$
If $(A,t_{10}),(B,t_7),(C,t_4)$ are independent,then $(A,{\overline{t}}_{10}),(B,{\overline{t}}_7),(C,{\overline{t}}_4),(\overline{A},t_{10}),(\overline{B},t_7),(\overline{C},t_4)$ will also be $independent.$
$\therefore$ The probability that she studies 4 hours per day given she is not successful,$P\left(C/\overline{S}\right)=\frac{P(C\cap \overline{S})}{P\left(\overline{S}\right)}$
$=\frac{0.42}{0.52}=\frac{21}{26}$