kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
\(? = 1 - P\left( {\underbrace {{\rm{all}}\,\,20\,\,{\rm{different}}\,\,{\rm{birthday}}\,\,{\rm{dates}}}_{{\rm{unfavorable}}}} \right)\)
\({\rm{Total}}:\,\,30 \cdot 30 \cdot \ldots \cdot 30 = {30^{20}}\,\,\,{\rm{equiprobable}}\,\,{\rm{possibilities}}\,\,\,\)
\({\rm{unfavorable}} = \,\,30 \cdot 29 \cdot \ldots \cdot 11\)
\(P\left( {{\rm{unfavorable}}} \right) = {{30 \cdot 29 \cdot \ldots \cdot 11} \over {{{30}^{20}}}} = 1 \cdot \underbrace {{{29} \over {30}} \cdot {{28} \over {30}} \cdot \ldots {{14} \over {30}}}_{ < < < \,\,1} \cdot \underbrace {{{13} \over {30}} \cdot {{12} \over {30}} \cdot {{11} \over {30}}}_{ \cong \,\,0.05} < < < 0.05 = 5\%\)
\(?\,\,\, > > > \,\,\,100\% - 5\% \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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37% (01:19) correct 
