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(B) Let $S = \{1, 2, \ldots, 100\}$. The total number of elements is $100$.Let $A$ be the event that the number is divisible by $2$. The elements of $

Vedclass · Accepted Answer

$\frac{23}{50}$

Vedclass · Answer

(B) Let $S = \{1, 2, \ldots, 100\}$. The total number of elements is $100$.Let $A$ be the event that the number is divisible by $2$. The elements of $A$ are $\{2, 4, 6, \ldots, 100\}$. The number of elements in $A$ is $n(A) = 50$.Let $B$ be the event that the number is divisible by $3$ or $5$. We want to find the conditional probability $P(B|A) = \frac{n(A \cap B)}{n(A)}$.$A \cap B$ is the set of numbers in $\{1, 2, \ldots, 100\}$ that are divisible by $2$ $AND$ (divisible by $3$ $OR$ divisible by $5$).This is equivalent to numbers divisible by $6$ or $10$.Numbers divisible by $6$ up to $100$: $\lfloor \frac{100}{6} \rfloor = 16$.Numbers divisible by $10$ up to $100$: $\lfloor \frac{100}{10} \rfloor = 10$.Numbers divisible by both $6$ and $10$ (i.e.,divisible by $30$): $\lfloor \frac{100}{30} \rfloor = 3$.By the Principle of Inclusion-Exclusion, $n(A \cap B) = 16 + 10 - 3 = 23$.The required probability is $\frac{n(A \cap B)}{n(A)} = \frac{23}{50}$.

$A$ number is selected at random from the set $\{1, 2, \ldots, 100\}$. Given that the selected number is divisible by $2$, what is the probability that it is also divisible by $3$ or $5$?

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