Let A and B be two independent events such th | vedclass

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

Question

(B) Given that $A$ and $B$ are independent events,we have $P(A \cap B) = P(A)P(B) = \frac{1}{12}$.Also,the probability that neither $A$ nor $B$ happen

Vedclass · Accepted Answer

$P(A) = \frac{1}{4}, P(B) = \frac{1}{3}$

Vedclass · Answer

(B) Given that $A$ and $B$ are independent events,we have $P(A \cap B) = P(A)P(B) = \frac{1}{12}$.Also,the probability that neither $A$ nor $B$ happens is $P(A^c \cap B^c) = P(A^c)P(B^c) = (1 - P(A))(1 - P(B)) = \frac{1}{2}$.Expanding the second equation: $1 - P(A) - P(B) + P(A)P(B) = \frac{1}{2}$.Substituting $P(A)P(B) = \frac{1}{12}$: $1 - (P(A) + P(B)) + \frac{1}{12} = \frac{1}{2}$.$P(A) + P(B) = 1 + \frac{1}{12} - \frac{1}{2} = \frac{12 + 1 - 6}{12} = \frac{7}{12}$.Let $x = P(A)$ and $y = P(B)$. We have $x + y = \frac{7}{12}$ and $xy = \frac{1}{12}$.The quadratic equation with roots $x$ and $y$ is $t^2 - (x+y)t + xy = 0$,which is $t^2 - \frac{7}{12}t + \frac{1}{12} = 0$.Multiplying by $12$: $12t^2 - 7t + 1 = 0$.$12t^2 - 4t - 3t + 1 = 0 \implies 4t(3t - 1) - 1(3t - 1) = 0$.$(4t - 1)(3t - 1) = 0$,so $t = \frac{1}{4}$ or $t = \frac{1}{3}$.Since $P(B) > P(A)$,we have $P(A) = \frac{1}{4}$ and $P(B) = \frac{1}{3}$.

Let $A$ and $B$ be two independent events such that $P(B) > P(A)$. If the probability that both $A$ and $B$ happen is $\frac{1}{12}$ and the probability that neither $A$ nor $B$ happens is $\frac{1}{2}$,then:

Similar Questions

If the integers $m$ and $n$ are chosen at random between $1$ and $100$,then the probability that a number of the form $7^m + 7^n$ is divisible by $5$ equals

$A$ and $B$ are independent events with $P(A)=\frac{3}{10}$ and $P(B)=\frac{2}{5}$. Then,the value of $P(A^{\prime} \cup B)$ is:

There are four fair dice $D_1, D_2, D_3$ and $D_4$. Each has six faces numbered $1, 2, 3, 4, 5$ and $6$. They are rolled one by one. What is the probability that the number shown on $D_4$ is equal to at least one of the numbers shown on $D_1, D_2$ and $D_3$ (in $/216$)?

If $E$ and $F$ are events with $P(E) \le P(F)$ and $P(E \cap F) > 0,$ then

Vedclass Test Series

Exam Paper Generator

Online Exam Module