Let B_{i} (i=1, 2, 3) be three independent ev | vedclass

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

Question

(B) Let $P(B_{1}) = p_{1}$,$P(B_{2}) = p_{2}$,and $P(B_{3}) = p_{3}$.Given that $p_{1}(1 - p_{2})(1 - p_{3}) = \alpha$ $...(i)$$p_{2}(1 - p_{1})(1 - p

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(B) Let $P(B_{1}) = p_{1}$,$P(B_{2}) = p_{2}$,and $P(B_{3}) = p_{3}$.Given that $p_{1}(1 - p_{2})(1 - p_{3}) = \alpha$ $...(i)$$p_{2}(1 - p_{1})(1 - p_{3}) = \beta$ $...(ii)$$p_{3}(1 - p_{1})(1 - p_{2}) = \gamma$ $...(iii)$And $(1 - p_{1})(1 - p_{2})(1 - p_{3}) = p$ $...(iv)$From these,we get $\frac{p_{1}}{1 - p_{1}} = \frac{\alpha}{p}$,$\frac{p_{2}}{1 - p_{2}} = \frac{\beta}{p}$,and $\frac{p_{3}}{1 - p_{3}} = \frac{\gamma}{p}$.Given equations: $(\alpha - 2\beta)p = \alpha\beta \Rightarrow \frac{1}{p} = \frac{1}{\beta} - \frac{2}{\alpha} \Rightarrow \frac{1}{\beta} = \frac{1}{p} + \frac{2}{\alpha}$.Also,$(\beta - 3\gamma)p = 2\beta\gamma \Rightarrow \frac{1}{2\gamma} - \frac{3}{2\beta} = \frac{1}{p} \Rightarrow \frac{1}{2\gamma} = \frac{1}{p} + \frac{3}{2\beta}$.Substituting $\frac{1}{\beta} = \frac{1}{p} + \frac{2}{\alpha}$ into the second equation:$\frac{1}{2\gamma} = \frac{1}{p} + \frac{3}{2}(\frac{1}{p} + \frac{2}{\alpha}) = \frac{1}{p} + \frac{3}{2p} + \frac{3}{\alpha} = \frac{5}{2p} + \frac{3}{\alpha}$.Using $\frac{\alpha}{p} = \frac{p_{1}}{1 - p_{1}}$ and $\frac{\gamma}{p} = \frac{p_{3}}{1 - p_{3}}$,we have $\frac{1}{2} \cdot \frac{1 - p_{3}}{p_{3}} = \frac{5}{2} + 3 \cdot \frac{1 - p_{1}}{p_{1}}$.Simplifying leads to $\frac{p_{1}}{p_{3}} = 6$.

Similar Questions

Two numbers are selected randomly from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ without replacement one by one. The probability that the minimum of the two numbers is divisible by $3$ or the maximum of the two numbers is divisible by $4$ is:

$A$ basket contains $5$ apples and $7$ oranges and another basket contains $4$ apples and $8$ oranges. One fruit is picked out from each basket. Find the probability that the fruits are both apples or both oranges.

You are given a box containing $20$ cards. Out of these,$10$ cards have the letter $I$ printed on them,and the other $10$ cards have the letter $T$ printed on them. If you draw three cards one after another with replacement,what is the probability of forming the word $IIT$?

Two squares are chosen at random on a chess-board. The probability that they have a side in common is

If two numbers $p$ and $q$ are chosen randomly from the set $\{1, 2, 3, 4\}$,one by one,with replacement,then the probability of getting $p^2 > 4q$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module