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 $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

There are $3$ houses available in a residential area. $3$ people apply for the houses. If each person applies for a house without consulting others,what is the probability that all $3$ apply for the same house (in $/9$)?

If $A$ and $B$ are two events,then the probability of the event that at most one of $A$ and $B$ occurs is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module