PROBABiLiTY

We chose the 2 persons in 23C₂ ways. They can have their birth day on any of the 365 days. The rest can have their birthday on the remaining 364.363.362....343 days. And the no. of total possible birthdates is 365²³.

So the required probability is 23C₂*365*364...*344365²³

0.5032(Approx.)

probability is 50%

\dpi{150} \fn_cs \textit{We try to find the probability that no 2 of the 23 people share the same birthday.}

\dpi{150} \fn_cs \textit{The first person can have his birthday on any of the 365 days.}
\dpi{150} \fn_cs \textit{The second person can have his birthday on any of the remaining 364 days.}
\dpi{150} \fn_cs \textit{...The }23^{rd}\textit{ person can have his birthday on any of the remaining 343 days.}

\dpi{150} \fn_cs \textit{Probability} =\frac{\textit{No. of favourable outcomes}}{\textit{ Total no. of possible outcomes}} = \frac{365\cdot 364\cdot 363\cdot ....\cdot 343}{365\cdot 365\cdot365\cdot ....\cdot 365 }

\dpi{150} \fn_cs \Rightarrow \textit{Probability}\approx 0.4968

\dpi{150} \fn_cs \therefore \textit{Probability of no 2 sharing birthdays = }1-0.4968 = 0.5032