JEE Main 2023 — Probability Question with Solution

The given probabilities for getting a head (H) and a tail (T) are as follows:

$$P(H)=\frac{3}{4},P(T)=\frac{1}{4}$$

The random variable X can take the values 1, 2, or 3. These correspond to the following events:

- X = 1 : A head is obtained on the first toss. This happens with probability $P(H)=\frac{3}{4}$.

- X = 2 : A tail is obtained on the first toss and a head on the second. This happens with probability $P(T)P(H)=\frac{1}{4}\times \frac{3}{4}$.

- X = 3 : Either two tails and then a head are obtained, or three tails are obtained.
This happens with probability $P(T)P(T)P(H)+P(T)P(T)P(T)={\left(\frac{1}{4}\right)}^2\times \frac{3}{4}+{\left(\frac{1}{4}\right)}^3$.

Now, we calculate the mean (expected value) of X:

$$\begin{array}{rl}E(X)& =1\cdot P(X=1)+2\cdot P(X=2)+3\cdot P(X=3)\\ & \\ & =1\cdot \frac{3}{4}+2\cdot \left(\frac{1}{4}\times \frac{3}{4}\right)+3\cdot \left[{\left(\frac{1}{4}\right)}^2\times \frac{3}{4}+{\left(\frac{1}{4}\right)}^3\right]\\ & \\ & =\frac{3}{4}+\frac{3}{8}+3\cdot \left(\frac{1}{64}+\frac{3}{64}\right)\\ & \\ & =\frac{3}{4}+\frac{3}{8}+\frac{3}{16}\\ & \\ & =3\cdot \left(\frac{7}{16}\right)\\ & \\ & =\frac{21}{16}.\end{array}$$