Probability: Basic JEE Main PYQs, 2020 to 2025

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

Two dice are thrown independently. Let $\mathrm{A}$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $\mathrm{B}$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and $\mathrm{C}$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then :

Let $\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$ be three mutually exclusive events such that $$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}$$ and $\mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1-\mathrm{p}}{2}$. If the maximum and minimum values of $\mathrm{p}$ are $\mathrm{p}_{1}$ and $\mathrm{p}_{2}$, then $\left(\mathrm{p}_{1}+\mathrm{p}_{2}\right)$ is equal to :