The Birthday Paradox

<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-the-birthday-paradox" target="_blank">The Birthday Paradox Exercise</a></p><p>Probability can be so counter-intuitive sometimes. It&#39;s like our brains don&#39;t process it well. Look up the Monty Hall Problem for a classic example of how tricky probability can be. Here&#39;s another famous example...</p> <p><strong><span style="color:#8e44ad;">The Birthday Paradox</span></strong></p> <p>At least how many random people do you need to throw in a room so that the probability of least one pair of people sharing a birthday is greater than $50\%$?</p> <p>Most people initially assume it&#39;s something like roughly &quot;half of the days in a year,&quot; so they guess something around $180$.&nbsp;</p> <p><strong><span style="color:#e74c3c;">The Answer is Actually $23$</span></strong></p> <p>What?! How? Check this out. Let&#39;s imagine we just have two random people in a room. What&#39;s the probability they don&#39;t share a birthday? For the first person, he or she can be born on any day, so that&#39;s $365$ possibilities. The second person can be born on all of those days except one, so that&#39;s $364$ possibilities. To calculate the probability, we do the following:</p> <p>$$\left(\frac{365}{365}\right)\left(\frac{364}{365}\right) \approx 99.7\%$$</p> <p><strong><span style="color:#27ae60;">What about three people not sharing a birthday?</span></strong></p> <p>Well the first person can be born on any of $365$ days, the second person on any of $364$ days, the third person on any of $363$ days, so the probability is...</p> <p>$$\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right) \approx 99.17\%$$</p> <p>If we continue...</p> <p style="text-align: center;">$5$ people not sharing: $\approx 97.3\%$</p> <p style="text-align: center;">$10$ people not sharing: $\approx 88.3\%$</p> <p style="text-align: center;">$15$ people not sharing: $\approx 74.7\%$</p> <p style="text-align: center;">$20$ people not sharing: $\approx 58.9\%$</p> <p style="text-align: center;">$23$ people not sharing: <span style="color:#27ae60;">$\approx 49.3\%$</span></p> <p>Wait, if the probability of $23$ random people not sharing a birthday is approximately $49.3\%$, that means the probability that at least two people share a birthday is $1$ minus this number.</p> <p style="text-align: center;"><span style="font-size:18px;">$$1 - 49.3\% = 50.7\%$$</span></p> <p>Mindblown.</p>