An integer is chosen at random from the integ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $S = \{1, 2, 3, \ldots, 50\}$,so the total number of outcomes is $n(S) = 50$.Let $A, B,$ and $C$ be the sets of multiples of $4, 6,$ and $7$ i

Vedclass · Accepted Answer

$\frac{21}{50}$

Vedclass · Answer

(B) Let $S = \{1, 2, 3, \ldots, 50\}$,so the total number of outcomes is $n(S) = 50$.Let $A, B,$ and $C$ be the sets of multiples of $4, 6,$ and $7$ in $S$ respectively.$A = \{4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48\} \implies n(A) = 12$.$B = \{6, 12, 18, 24, 30, 36, 42, 48\} \implies n(B) = 8$.$C = \{7, 14, 21, 28, 35, 42, 49\} \implies n(C) = 7$.Now,find the intersections:$A \cap B = \{12, 24, 36, 48\} \implies n(A \cap B) = 4$.$B \cap C = \{42\} \implies n(B \cap C) = 1$.$A \cap C = \{28\} \implies n(A \cap C) = 1$.$A \cap B \cap C = \emptyset \implies n(A \cap B \cap C) = 0$.Using the inclusion-exclusion principle:$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$.$n(A \cup B \cup C) = 12 + 8 + 7 - 4 - 1 - 1 + 0 = 21$.The required probability is $\frac{n(A \cup B \cup C)}{n(S)} = \frac{21}{50}$.

An integer is chosen at random from the integers $\{1, 2, 3, \ldots, 50\}$. The probability that the chosen integer is a multiple of at least one of $4, 6,$ and $7$ is

Similar Questions

If $U$ is the universal set with $100$ elements; $A$ and $B$ are two sets such that $n(A)=50$,$n(B)=60$,and $n(A \cap B)=20$,then find $n(A^{\prime} \cap B^{\prime})$.

Let $S = \{4, 6, 9\}$ and $T = \{9, 10, 11, \ldots, 1000\}$. If $A = \{a_{1} + a_{2} + \ldots + a_{k} : k \in N, a_{1}, a_{2}, \ldots, a_{k} \in S\}$,then the sum of all the elements in the set $T - A$ is equal to:

The probability of choosing at random a number that is divisible by $6$ or $8$ from among $1$ to $90$ is equal to

The lengths of the sides and the diagonal of an isosceles trapezium form a two-element set $\{a, b\}$. If $a > b$,then $a / b$ equals

In a group of students,$100$ students know Hindi,$50$ know English,and $25$ know both. Each student knows either Hindi or English. How many students are there in the group?

Vedclass Test Series

Exam Paper Generator

Online Exam Module