Let there be three independent events E_{1}, | vedclass

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

Question

(B) Let $P(E_{1}) = P_{1}$,$P(E_{2}) = P_{2}$,and $P(E_{3}) = P_{3}$.Since the events are independent,the probability that only $E_{1}$ occurs is $\al

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(B) Let $P(E_{1}) = P_{1}$,$P(E_{2}) = P_{2}$,and $P(E_{3}) = P_{3}$.Since the events are independent,the probability that only $E_{1}$ occurs is $\alpha = P_{1}(1 - P_{2})(1 - P_{3})$.The probability that only $E_{2}$ occurs is $\beta = (1 - P_{1})P_{2}(1 - P_{3})$.The probability that only $E_{3}$ occurs is $\gamma = (1 - P_{1})(1 - P_{2})P_{3}$.The probability that none of the events occur is $p = (1 - P_{1})(1 - P_{2})(1 - P_{3})$.Given $(\alpha - 2\beta)p = \alpha\beta$,we substitute the expressions:$\left(P_{1}(1 - P_{2})(1 - P_{3}) - 2(1 - P_{1})P_{2}(1 - P_{3})\right)p = P_{1}(1 - P_{2})(1 - P_{3}) \cdot (1 - P_{1})P_{2}(1 - P_{3})$.Dividing both sides by $(1 - P_{1})(1 - P_{2})(1 - P_{3})^2$,we get:$\frac{P_{1}}{1 - P_{1}} - \frac{2P_{2}}{1 - P_{2}} = \frac{P_{1}P_{2}}{(1 - P_{1})(1 - P_{2})}$.This simplifies to $P_{1} = 2P_{2}$.Similarly,from $(\beta - 3\gamma)p = 2\beta\gamma$,we get $P_{2} = 3P_{3}$.Therefore,$P_{1} = 2(3P_{3}) = 6P_{3}$.Thus,$\frac{P_{1}}{P_{3}} = 6$.

Similar Questions

In a random experiment,a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

Two coins are tossed. Let $A$ be the event that the first coin shows a head and $B$ be the event that the second coin shows a tail. The two events $A$ and $B$ are:

Given two independent events,if the probability that exactly one of them occurs is $\frac{26}{49}$ and the probability that none of them occurs is $\frac{15}{49}$,then the probability of the more probable of the two events is (in $/7$)

If $E_1, E_2, \ldots, E_n$ are independent events such that $P(E_r) = \frac{1}{1+r}$ for $r = 1, 2, \ldots, n$,then the probability that at least one of $E_1, E_2, \ldots, E_n$ happens is

Vedclass Test Series

Exam Paper Generator

Online Exam Module