JEE Advanced · Mathematics · 32. Probability

Paragraph:

A box $B_{1}$ contains $1$ white ball, $3$ red balls and $2$ black balls. Another box $B_{2}$ contains $2$ white balls, $3$ red balls and $4$ black balls. A third box $B_{3}$ contains $3$ white balls, $4$ red balls and $5$ black balls.

Question:

If $1$ ball is drawn from each of the boxes $B_{1}, B_{2}$ and $B_{3}$, the probability that all $3$ drawn balls are of the same colour is

A $\frac{82}{648}$
B $\frac{90}{648}$
C $\frac{558}{648}$
D $\frac{566}{648}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{82}{648}$

Step-by-step Solution

Detailed explanation

P (W W W) + P (R R R) + P (B B B)

(\frac{1}{6} \times \frac{2}{9} \times \frac{3}{12}) + (\frac{3}{6} \times \frac{3}{9} \times \frac{4}{12}) + (\frac{2}{6} \times \frac{4}{9} \times \frac{5}{12})

\Rightarrow \frac{6 + 36 + 40}{6 \times 9 \times 12} \Rightarrow \frac{82}{648}

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.

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	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)