JEE Main 2022 — Probability Question with Solution

Student guesses only two wrong. So there are three possibilities.

(i) Student guesses both wrong from 1^st section

(ii) Student guesses both wrong from 2^nd section

(iii) Student guesses two wrong one from each section

Required probabilities

$$={}^4{C}_2{\left(\frac{3}{4}\right)}^2{\left(\frac{1}{4}\right)}^2{\left(\frac{1}{6}\right)}^6+{}^6{C}_2{\left(\frac{3}{4}\right)}^2{\left(\frac{1}{4}\right)}^4{\left(\frac{3}{4}\right)}^4+{}^4{C}_1.{}^6{C}_1\left(\frac{3}{4}\right)\left(\frac{1}{4}\right){\left(\frac{3}{4}\right)}^3{\left(\frac{1}{4}\right)}^5$$

$$=\frac{1}{4^{10}}\left[6\times 9+15\times 9^4+24\times 9^2\right]$$

$$=\frac{27}{4^{10}}\left[2+27\times 15+72\right]$$

$$=\frac{27\times 479}{4^{10}}$$