Birthday paradox
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The birthday paradox states that in a random group of 23 people, there is about a 50% chance that two people will have the same birthday. It seems counterintuitive because we expect the probability of a match to be low when comparing individual birthdays. However, with 23 people there are 253 possible pairs to compare, and taking into account all the combinations, the odds of at least one match is actually around 50%. The probability increases rapidly with larger groups, reaching over 99% for a group of 75 people.
Birthday paradox
- 1. Birthday Paradox Dr. Sajid Iqbal
- 2. 2 Today, we hope to learn ... Birthday paradox
- 3. 3 What is Birthday paradox? The Birthday Paradox, aka the Birthday Problem, states that in a random group of 23 people, there is about a 50 % chance that two people have the same birthday. The birthday paradox is strange, counter- intuitive, and completely true.
- 4. 4 What is Birthday paradox?(contd.) It’s only a paradox because our brains can’t handle the compounding power of exponents. We expect probabilities to be linear and only consider the scenarios we’re involved in (both faulty assumptions).
- 5. 5 It is used in several different areas e.g., cryptography and hashing algorithms. The reason this is so surprising is because we are used to comparing our particular birthdays with others. For instance, if you meet someone randomly and ask him what his birthday is, the chance of the two of you having the same birthday is only 1/365 (0.27%). What is Birthday paradox?(contd.)
- 6. 6 Birthday paradox (contd.) A person's birthday is one out of 365 possibilities (excluding Feb 29 birthdays). In other words, the probability of any two individuals having the same birthday is extremely low. Even if you ask 20 people, the probability is still low -- less than 5%.
- 7. 7 Birthday paradox (contd.) BUT, in a group of 23 people, there are (22 + 21 + 20 + 19 + … +1) comparisons, or 253combinations, that can be made. So, we're not looking at just one comparison, but at 253 comparisons.
- 8. 8 Birthday paradox (contd.) Every one of the 253 combinations has the same odds, 99.7 %, of not being a match. If you multiply 99.7 percent by 99.7 253 times, or calculate (364/365)253, you'll find there's a 49.952 % chance that all 253 comparisons contain no matches. Consequently, the odds that there is a birthday match in those 253 comparisons is 1 – 49.952 percent = 50.048 %, or just over half! The more trials you run, the closer the actual probability should approach 50 percent.
- 9. 9 Birthday paradox (contd.) In a room of just 23 people, there is a 50-50 chance of two people having the same birthday. In a room of 75 there’s a 99.9% likelihood of two people matching.