Q.A six-sided die is marked 2' on one face, 3' on two of its faces, and `4' on the remaining three faces. The die is thrown twice. If $X$ denotes the total score in the two throws, find the values of the random variable and the number of points in its inverse images.

Lists all 8 equally likely outcomes of tossing 3 coins, groups them by number of tails, and counts each group's size.

1. When 3 fair coins are tossed simultaneously, the sample space has $2^3=8$ equally likely outcomes: $S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$.
2. Let $X$ = number of tails in an outcome, so $X:S\to\mathbb{R}$.
3. $X=0$: only $HHH$ — $1$ point.
4. $X=1$: $HHT,HTH,THH$ — $3$ points.
5. $X=2$: $HTT,THT,TTH$ — $3$ points.
6. $X=3$: only $TTT$ — $1$ point. …

A continuous random variable takes every value in an interval, so probabilities come from areas under a probability density function $f(x)$, and the probability of any single exact value is zero.

A random variable is continuous when it can assume any value within a range (an interval of the real line) — for example the exact weight of bread sold or the exact life of a bulb. Its key properties are:

1. Range is an interval (uncountable). $X$ can take infinitely many values in $[a, b]$ (or over all of $\mathbb{R}$).
2. Point probability is zero. For any single value $x$, $P(X = x) = 0$; consequently $P(a \le X \le b) = P(a < X < b)$.
3. Described by a density $f(x)$ (the probability density function) satisfying $f(x) \ge 0 \quad\text{and}\quad \int_{-\infty}^{\infty} f(x)\,dx = 1.$
4. Probabilities are areas. $P(a \le X \le b) = \int_a^b f(x)\,dx$, the area under the curve $y = f(x)$ between $x = a$ and $x = b$. …