If the events A and B are mutually exclusive | vedclass

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

Question

(C) For any event $E$,the probability $P(E)$ must satisfy $0 \le P(E) \le 1$. For event $A$: $0 \le \frac{3x+1}{3} \le 1 \Rightarrow 0 \le 3x+1 \le 3

Vedclass · Accepted Answer

$[-\frac{1}{3}, \frac{5}{9}]$

Vedclass · Answer

(C) For any event $E$,the probability $P(E)$ must satisfy $0 \le P(E) \le 1$. For event $A$: $0 \le \frac{3x+1}{3} \le 1 \Rightarrow 0 \le 3x+1 \le 3 \Rightarrow -1 \le 3x \le 2 \Rightarrow -\frac{1}{3} \le x \le \frac{2}{3}$. For event $B$: $0 \le \frac{1-x}{4} \le 1 \Rightarrow 0 \le 1-x \le 4 \Rightarrow -1 \le -x \le 3 \Rightarrow -3 \le x \le 1$. Since $A$ and $B$ are mutually exclusive,$P(A \cup B) = P(A) + P(B) \le 1$. $\frac{3x+1}{3} + \frac{1-x}{4} \le 1 \Rightarrow \frac{4(3x+1) + 3(1-x)}{12} \le 1 \Rightarrow 12x + 4 + 3 - 3x \le 12 \Rightarrow 9x + 7 \le 12 \Rightarrow 9x \le 5 \Rightarrow x \le \frac{5}{9}$. Combining all conditions: $x \ge -\frac{1}{3}$,$x \le \frac{2}{3}$,$x \ge -3$,$x \le 1$,and $x \le \frac{5}{9}$. The intersection of these intervals is $[-\frac{1}{3}, \frac{5}{9}]$.

If the events $A$ and $B$ are mutually exclusive events such that $P(A) = \frac{3x + 1}{3}$ and $P(B) = \frac{1 - x}{4}$,then the set of possible values of $x$ lies in the interval

Similar Questions

An experiment consists of recording the boy-girl composition of families with $2$ children. What is the sample space if we are interested in the number of girls in the family?

$A$ box contains $100$ tickets numbered $1$ to $100$. $A$ ticket is drawn at random from the box. Then the probability that the number on the ticket is a perfect square is

Out of the given $25$ consecutive positive integers, three integers are drawn. If the least integer among the given $25$ integers is an odd number, then the probability that the sum of the three integers drawn is an even number is

Out of $100$ students,two sections of $40$ and $60$ are formed. If you and your friend are among the $100$ students,what is the probability that you both enter different sections?

$A$ coin is tossed. If the outcome is a head,a die is thrown. If the die shows up an even number,the die is thrown again. What is the sample space for the experiment?

Vedclass Test Series

Exam Paper Generator

Online Exam Module