JEE Main 2023 — Probability Question with Solution

Let $X$ be the number rolled on the die, and let $W$ be the event that all balls drawn are white. We want to find the probability $P(W)$, which can be calculated using the law of total probability as follows :

$$P(W)=\sum _{x=1}^{6}P(W|X=x)P(X=x)$$

The probability of rolling any number from 1 to 6 on the die is equal, so $P(X=x)=\frac{1}{6}$ for all $x\in \{1,2,3,4,5,6\}$.

Now let's calculate the conditional probabilities $P(W|X=x)$ for each possible value of $x$ :

1. $P(W|X=1)=\frac{{}^6{C}_1}{{}^{10}{C}_1}=\frac{6}{10}=\frac{3}{5}$, since there are 6 white balls out of a total of 10 balls.

2. $P(W|X=2)=\frac{{}^6{C}_2}{{}^{10}{C}_2}=\frac{15}{45}=\frac{1}{3}$, since there are 15 ways to choose 2 white balls out of 6, and 45 ways to choose 2 balls out of 10.

3. $P(W|X=3)=\frac{{}^6{C}_3}{{}^{10}{C}_3}=\frac{20}{120}=\frac{1}{6}$, since there are 20 ways to choose 3 white balls out of 6, and 120 ways to choose 3 balls out of 10.

4. $P(W|X=4)=\frac{{}^6{C}_4}{{}^{10}{C}_4}=\frac{15}{210}=\frac{1}{14}$, since there are 15 ways to choose 4 white balls out of 6, and 210 ways to choose 4 balls out of 10.

5. $P(W|X=5)=\frac{{}^6{C}_5}{{}^{10}{C}_5}=\frac{6}{252}=\frac{1}{42}$, since there are 6 ways to choose 5 white balls out of 6, and 252 ways to choose 5 balls out of 10.

6. $P(W|X=6)=\frac{{}^6{C}_6}{{}^{10}{C}_6}=\frac{1}{210}$, since there are 1 ways to choose 6 white balls out of 6, and 210 ways to choose 6 balls out of 10.

Using the law of total probability, we have :

$$P(W)=\frac{1}{6}\left(P(W|X=1)+P(W|X=2)+P(W|X=3)+P(W|X=4)+P(W|X=5)+P(W|X=6)\right)$$

$$P(W)=\frac{1}{6}\left(\frac{3}{5}+\frac{1}{3}+\frac{1}{6}+\frac{1}{14}+\frac{1}{42}+\frac{1}{210}\right)$$

To simplify this expression, find a common denominator :

$$P(W)=\frac{1}{6}\left(\frac{126}{210}+\frac{70}{210}+\frac{35}{210}+\frac{15}{210}+\frac{5}{210}+\frac{1}{210}\right)$$

Add the fractions :

$$P(W)=\frac{1}{6}\left(\frac{126+70+35+15+5+1}{210}\right)=\frac{42}{210}=\frac{1}{5}$$