JEE Advanced · Mathematics · 32. Probability

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station $A$ and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station $B$ is green, then the probability that the original signal green is

A
$\frac{3}{5}$
B
$\frac{6}{7}$
C
$\frac{20}{23}$
D
$\frac{9}{20}$

Verified Solution

Answer & Solution

Correct Answer

(C)
$\frac{20}{23}$

Step-by-step Solution

Detailed explanation

From the tree-diagram it follows that

\[
\begin{gathered}
P\left(B_G \mid G\right)=\frac{10}{16}=\frac{5}{8} \\
\therefore \quad P\left(B_G \cap G\right)=\frac{5}{8} \times \frac{4}{5}=\frac{1}{2} \\
P\left(G \mid B_G\right)=\frac{\frac{1}{2}}{P\left(B_G\right)}=\frac{1}{2} \times \frac{80}{46}=\frac{20}{23}
\end{gathered}
\]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

let

f_{1} : R \to R, f_{2} : [0, \infty) \to R, f_{3} : R \to R a n d f_{4} : R \to [0, \infty)

be defined by

f_{1} (x) = \{\begin{matrix} |x| & i f & x < 0 \\ e^{x} & i f & x \geq 0 \end{matrix}; f_{2} (x) = x^{2}; f_{3} (x) = \{\begin{matrix} \sin x & i f & x < 0 \\ x & i f & x \geq 0 \end{matrix} a n d f_{4} (x) = \{\begin{matrix} f_{2} (f_{1} (x)) & i f & x < 0 \\ f_{2} (f_{1} (x)) - 1 & i f & x \geq 0 \end{matrix}

	List – I		List – II
(A)	$f_{4}$ is	(P)	Onto but not one – one
(B)	$f_{3}$ is	(Q)	neither continuous nor one-one
(C)	$f_{2}$ of $f_{1}$ is	(R)	differentiable but not one-one
(D)	$f_{2}$ is	(S)	continuous and one-one

JEE Advanced 2014 Medium

Let $\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.$ then $\int$ is

JEE Advanced 2012 Medium

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