Same Day Birthday-A Paradox?
The excitement of this occasion doubles when one shares his birthday with another person. In this regard the Birthday paradox has a major role to play. The birthday paradox states that given a group of 23 randomly chosen people, the probability is more than 50% that at least two of them will have the same birthday. If the number of people increases to 60 or more, the probability is greater than 99%. However it cannot actually be 100% unless there are at least 366 people. One should not take it to be a paradox in the true sense of the word , as in the sense of leading to a logical contradiction. In fact it is described as a paradox because mathematical truth contradicts candid or gullible intuition.
One can try it himself. If one is at a gathering of 20 or 30 people, and each individual’s date of birth is asked, it is likely that two people in the group will have the same date of birth. It always surprises people! The reason this is so surprising is because an individual is used to comparing his particular birthdays with others. For example, if a person meets someone randomly and asks him his date of birth, the chance of the two of them having the same birthday is only 1/365 (0.27%) which is extremely low. Even if he asks 20 people, the probability is still low — less than 5%. So one feels that it is very rare to meet anyone with the same date of birth as his.
When 20 people are put in a room, however, the thing that changes is the fact that each of the 20 people is now asking each of the other 19 people about their date of birth. Each individual person only has a small, less than 5%, chance of success, but each person is trying it 19 times. So that increases the probability dramatically. If one wants to calculate the exact probability, one way to look at it is like this. He should mark his birthday on the calendar. The next person who walks in has only a 364 possible open days available, so the probability of the two dates not colliding is 364/365. The next person has only 363 open days, so the probability of not colliding is 363/365. If one multiplies the probabilities for all 20 people not colliding, then one gets: 364/365 * 363/365 * … 365-20+1/365 = Chances of no collisions. That is the probability of no collisions, so the probability of collisions is 1 minus that number. The next time you are with a group of 30 people, try it!
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