Jars P, Q, and R each contain marbles of various colors. One marble will be selected at random from each jar. If the chance that all 3 marbles selected are blue is 1/8 , what is the chance that at least 1 of the marbles selected is blue?
(1) The chance that none of the 3 marbles selected is blue is 7/80
(2) The chance that at least 1 of the marbles selected from jars P and Q is blue is 23/30
Dear varun289 and tapojoy, this is a beautiful question, although it is out-of-GMAT´s scope.
(The solution will be brief, although the rationale to find the BIFURCATION is really interesting.)
(I have changed the statements order, just because it would be nicer to type less first.)
Let "yesP" denote the extraction of a blue marble of jar P. (The other notations are obvious from that.)
\[P\left( {{\text{yesP}}\,\,,\,\,{\text{yesQ}}\,\,,\,\,{\text{yesR}}} \right) = \frac{1}{8}\]
\[? = 1 - P\left( {{\text{noP}}\,,\,\,{\text{noQ}}\,,{\text{noR}}} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{? = P\left( {{\text{noP}}\,,\,\,{\text{noQ}}\,,{\text{noR}}} \right)}\]
\[\left( 1 \right)\,\,\frac{7}{{80}} = P\left( {{\text{noP}}\,,\,\,{\text{noQ}}\,,{\text{noR}}} \right)\,\,\, = \,\,\,?\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}{\text{.}}\]
\[\left( 2 \right)\,\,\left\{ \begin{gathered}
\,\,\left( {2a} \right)\,\,If\,\,P\,\,:\,\,5\,{\text{marbles}}\,{\text{,}}\,\,{\text{3}}\,\,{\text{blue}}\,\,{\text{/}}\,\,\,Q\,\,:\,\,12\,{\text{marbles}}\,{\text{,}}\,\,{\text{5}}\,\,{\text{blue}}\,\,\,{\text{/}}\,\,\,{\text{R}}\,\,{\text{:}}\,\,{\text{2}}\,\,{\text{marbles}}\,\,{\text{,}}\,\,{\text{1}}\,\,{\text{blue}}\,\,\,\,\, \Rightarrow \,\,\,\,? = \,\,\frac{7}{{60}} \hfill \\
\,\,\left( {2b} \right)\,\,If\,\,P\,\,:\,\,3\,{\text{marbles}}\,{\text{,}}\,\,{\text{1}}\,\,{\text{blue}}\,\,{\text{/}}\,\,\,Q\,\,:\,\,20\,{\text{marbles}}\,{\text{,}}\,\,{\text{13}}\,\,{\text{blue}}\,\,\,{\text{/}}\,\,\,{\text{R}}\,\,{\text{:}}\,\,{\text{26}}\,\,{\text{marbles}}\,\,{\text{,}}\,\,{\text{15}}\,\,{\text{blue}}\,\,\,\,\, \Rightarrow \,\,\,\,? = \,\,\frac{{77}}{{780}} \ne \frac{7}{{60}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}.\]
\[\left( {2a} \right)\,\,\,\left\{ \begin{gathered}
\,P\left( {{\text{yesP}}\,\,,\,\,{\text{yesQ}}\,\,,\,\,{\text{yesR}}} \right) = \frac{3}{5} \cdot \frac{5}{{12}} \cdot \frac{1}{2} = \frac{1}{8}\,\,\,\,\,\,{\text{good!}} \hfill \\
\,P\left( {{\text{noP}}\,\,,\,\,{\text{noQ}}\,} \right) = \,\,\frac{2}{5} \cdot \frac{7}{{12}} = \frac{7}{{30}}\,\,\,\,\,\,{\text{good!}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {2a} \right)\,\,\,{\text{viable}}\]
\[\left( {2b} \right)\,\,\,\left\{ \begin{gathered}
\,P\left( {{\text{yesP}}\,\,,\,\,{\text{yesQ}}\,\,,\,\,{\text{yesR}}} \right) = \frac{1}{3} \cdot \frac{{13}}{{20}} \cdot \frac{{15}}{{26}} = \frac{1}{8}\,\,\,\,\,\,{\text{good!}} \hfill \\
\,P\left( {{\text{noP}}\,\,,\,\,{\text{noQ}}\,} \right) = \,\,\frac{2}{3} \cdot \frac{7}{{20}} = \frac{7}{{30}}\,\,\,\,\,\,{\text{good!}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {2b} \right)\,\,\,{\text{viable}}\]
The mathematical maturity needed to deal with this sort of problem, but in GMAT´s reality, is developed in our course.
Regards,
fskilnik.
