GATE CS 1995 Q1.18 — Probability: Random Variables and Distributions

To find the probability that a randomly selected 3-digit number between $100$ and $999$ (both inclusive) does not contain the digit $7$:

Step 1: Find the total number of possible outcomes
The numbers from $100$ to $999$ inclusive are all the $3$-digit numbers.
$$\text{Total numbers} = 999 - 100 + 1 = 900$$

Step 2: Find the number of favorable outcomes (numbers without the digit $7$)
Let a $3$-digit number be represented as $d_1 d_2 d_3$:

• The hundreds digit ($d_1$) cannot be $0$ or $7$. Thus, $d_1 \in \{1, 2, 3, 4, 5, 6, 8, 9\}$, giving $8$ choices.
• The tens digit ($d_2$) cannot be $7$. Thus, $d_2 \in \{0, 1, 2, 3, 4, 5, 6, 8, 9\}$, giving $9$ choices.
• The units digit ($d_3$) cannot be $7$. Thus, $d_3 \in \{0, 1, 2, 3, 4, 5, 6, 8, 9\}$, giving $9$ choices.

Step 3: Calculate the probability
$$\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{648}{900}$$ Simplifying the fraction by dividing both the numerator and denominator by $36$: $$\text{Probability} = \frac{18}{25}$$