GATE CS 2014 Set 2 Q48 — Probability: Random Variables and Distributions

Let the sample space be $S = \{1, 2, 3, \dots, 100\}$. The total number of elements is $|S| = 100$.

We want to find the probability that a randomly chosen integer from $S$ is not divisible by $2$, $3$, or $5$.

Let us define the following sets of integers in $S$:

• $A$: Divisible by $2$
• $B$: Divisible by $3$
• $C$: Divisible by $5$

Using the Principle of Inclusion-Exclusion (PIE), the number of integers divisible by at least one of $2$, $3$, or $5$ is given by:

Let's calculate each term:

• $|A| = \lfloor \frac{100}{2} \rfloor = 50$
• $|B| = \lfloor \frac{100}{3} \rfloor = 33$
• $|C| = \lfloor \frac{100}{5} \rfloor = 20$
• $|A \cap B|$ (divisible by $\text{lcm}(2,3) = 6$): $\lfloor \frac{100}{6} \rfloor = 16$
• $|B \cap C|$ (divisible by $\text{lcm}(3,5) = 15$): $\lfloor \frac{100}{15} \rfloor = 6$
• $|C \cap A|$ (divisible by $\text{lcm}(5,2) = 10$): $\lfloor \frac{100}{10} \rfloor = 10$
• $|A \cap B \cap C|$ (divisible by $\text{lcm}(2,3,5) = 30$): $\lfloor \frac{100}{30} \rfloor = 3$

Substituting these values into the PIE formula:

The number of integers in $S$ that are not divisible by $2$, $3$, or $5$ is:

Thus, the required probability is: