JEE Main 2013 — Mathematics Probability & Statistics

Let the probability of occurrence of first event A, be ' $a$a ' i..e., $\mathrm{P}(\mathrm{A})=\mathrm{a}$P(A)=a $$\therefore \mathrm{P}(\text { not } \mathrm{A})=1-a$$∴P( not A)=1−a And also suppose that probability of occurrence of second event $\mathrm{B}, \mathrm{P}(\mathrm{B})=b$B,P(B)=b, $$\therefore \mathrm{P}(\operatorname{not} \mathrm{B})=1-b$$∴P(notB)=1−b Now, $\mathrm{P}(\mathrm{A}$P(A and not $\mathrm{B})+\mathrm{P}($B)+P( not $\mathrm{A}$A and $\mathrm{B})=\frac{26}{49}$B)=4926​ $$\begin{aligned} & \Rightarrow \quad \mathrm{P}(\mathrm{A}) \times \mathrm{P}(\operatorname{not} \mathrm{B})+\mathrm{P}(\operatorname{not} \mathrm{A}) \times \mathrm{P}(\mathrm{B})=\frac{26}{49} \\ & \Rightarrow \quad a \times(1-b)+(1-a) b=\frac{26}{49} \\ & \Rightarrow \quad a+b-2 a b=\frac{26}{49} \end{aligned}$$​⇒P(A)×P(notB)+P(notA)×P(B)=4926​⇒a×(1−b)+(1−a)b=4926​⇒a+b−2ab=4926​​ And $\mathrm{P}($P( not $\mathrm{A}$A and not $\mathrm{B})=\frac{15}{49}$B)=4915​ $$\begin{aligned} & \Rightarrow \quad \mathrm{P}(\operatorname{not} \mathrm{A}) \times \mathrm{P}(\text { not } \mathrm{B})=\frac{15}{49} \\ & \Rightarrow \quad(1-a) \times(1-b)=\frac{15}{49} \\ & \Rightarrow 1-\mathrm{b}-\mathrm{a}+\mathrm{ab}=\frac{15}{49} \\ & \Rightarrow a+b-a b=\frac{34}{49} \end{aligned}$$​⇒P(notA)×P( not B)=4915​⇒(1−a)×(1−b)=4915​⇒1−b−a+ab=4915​⇒a+b−ab=4934​​ From (i) and (ii), $$a+b=\frac{42}{49}$$a+b=4942​ and $a b=\frac{8}{49}$ab=498​ $$\begin{aligned} & (a-b)^2=(a+b)^2-4 a b=\frac{42}{49} \times \frac{42}{49}-\frac{4 \times 8}{49} \\ & =\frac{196}{2401} \end{aligned}$$​(a−b)2=(a+b)2−4ab=4942​×4942​−494×8​=2401196​​ $$\therefore \quad a-b=\frac{14}{49}$$∴a−b=4914​ From (iii) and (iv), $$a=\frac{4}{7}, b=\frac{2}{7}$$a=74​,b=72​ Hence probability of more probable of the $$\text { two events }=\frac{4}{7}$$ two events =74​