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The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 6 \cdot 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 6 \cdot 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 83.13$. Hence we would expect to need $83$ or $84$ people in the room before having a 50-50 chance of getting three people with the same birthday.

UPDATE: This is only an approximation, and it turns out it's off by more than I expected. To get an exact answer, I used the exact formula given here, and found that $N = 87$ is actually closest, with a probability of $.49945$ that three people have the same birthday.

The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 6 \cdot 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 6 \cdot 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 83.13$. Hence we would expect to need $83$ or $84$ people in the room before having a 50-50 chance of getting three people with the same birthday.

UPDATE: This is only an approximation, and it turns out it's off by more than I expected. To get an exact answer, I used the exact formula given here, and found that $N = 87$ is actually closest, with a probability of $.49945$ that three people have the same birthday.

The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 6 \cdot 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 6 \cdot 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 83.13$. Hence we would expect to need $83$ or $84$ people in the room before having a 50-50 chance of getting three people with the same birthday.

The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 6 \cdot 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 6 \cdot 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 83.13$. Hence we would expect to need $83$ or $84$ people in the room before having a 50-50 chance of getting three people with the same birthday.

UPDATE: This is only an approximation, and it turns out it's off by more than I expected. To get an exact answer, I used the exact formula given here, and found that $N = 87$ is actually closest, with a probability of $.49945$ that three people have the same birthday.

The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 46.2$. Hence we would expect to need $46$ or $47$ people in the room before having a 50-50 chance of getting three people with the same birthday.

The question linked in the comments, while not your exact question, is very relevant. In particular, reading the answers posted there will tell you that an exact computation for triples is gross. However, Byron Schmuland's answer gives us a useful estimate: $$ P(\text{at least one triple with } N \text{ people}) \approx 1 - e^{-{N \choose 3}/365^2} $$ From this we can simply solve for what $N$ makes this value $\frac12$. We need \begin{align*} 1 - e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{-{N \choose 3}/365^2} &= \frac12 \\ e^{{N \choose 3}/365^2} &= 2 \\ {N \choose 3} &= 365^2 \ln 2 \\ N(N-1)(N-2) &= 6 \cdot 365^2 \ln 2 \\ N^3 - 3N^2 + 2N - 6 \cdot 365^2 \ln 2 &= 0 \end{align*}

A computer calculation reveals that the only real root of this cubic polynomial is $N \approx 83.13$. Hence we would expect to need $83$ or $84$ people in the room before having a 50-50 chance of getting three people with the same birthday.