If three person are in a room, what is the probability that for two of them to be born the same day of the week?
This was a question asked at an IMC Financial Market job interview.
I know that the probability for someone to be born the same day of the week with another one is the probability to be born the same day of the week with the first people $P_1$ and the probability not to be born tthe same day of the week with $P_1$ but with $P_2$.
Therefore, the probability $P(X)$ for all of them to be born the same day is:
\begin{equation}
P(X)=\frac{1}{21}
\end{equation}
Therfore the prbability for none of them to be born the same day of the week is
\begin{equation}
P(\bar X)=\frac{20}{21}
\end{equation}
Therefore the probability for at least two of them to be born the same day of the week is:
\begin{equation}
P(X_2)=\frac{20}{21}-2*\frac{1}{7}
\end{equation}
Am I right?
Best Answer
Start with any one of the three. The chances that the 2nd does not share his birthday is $\frac{6}{7}$. The chances that the 3rd does not share with either is $\frac{5}{7}$. Therefore the odds of a match are: $$1-\frac{6}{7}*\frac{5}{7}=1-\frac{30}{49}=\frac{19}{49}$$