When comparing probabilities with birthdays, it can be easier to look at the probability that people do not share a birthday. A person's birthday is one out of 365 possibilities (excluding February 29 birthdays). The probability that a person does not have the same birthday as another person is 364 divided by 365 because there are 364 days that are not a person's birthday. This means that any two people have a 364/365, or 99.726027 percent, chance of not matching birthdays. Show As mentioned before, in a group of 23 people, there are 253 comparisons, or combinations, that can be made. So, we're not looking at just one comparison, but at 253 comparisons. Every one of the 253 combinations has the same odds, 99.726027 percent, of not being a match. If you multiply 99.726027 percent by 99.726027 253 times, or calculate (364/365)253, you'll find there's a 49.952 percent 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 percent, or just over half! The more trials you run, the closer the actual probability should approach 50 percent. I was born on the 2nd of August, exactly 33 years before my father was born. I always taught the fact of sharing the birthday with my dad was something really unique. I don’t even have two friends who were born on the same day. I never really thought about the math of two people having the same birthday. If one day a friend of mine hadn’t talked to me about the birthday paradox, I probably never would. He said, “You're at a party. There are exactly 23 people in the room, what are the odds of two people sharing the same birthday?” I said, “I don’t know but I think they are pretty low.” “Actually, it is more than 50%.” “Shut up! No way that’s true!” As it turns out, it is. Introduction To solve this problem we have to answer a simple question:
Before we begin we have to make some assumptions. First, we consider a year of 365 days (no leap years, sorry). This means that to have a 100% probability we need 366 people. The second assumption is that all the 365 birthdays are equally likely. In reality, this is not true but the results are affected only slightly. Actually, this is the worst case (learn more know here) Solution Let’s start simple. What’s the chance that two people share the same birthday? The first person can be born on any day of the year, this means that the probability is 365/365 = 1. The second person has to be born on the same day as the first and there is a 1/365 chance of that happening. These two events need to happen at the same time so the probability is: Not very high as we expected. Now we can consider a group of three people (let’s call them A, B, C). To know the probability of at least two people sharing their birthday we have to calculate:
This is not too difficult. However, if we want to do the same for a larger number of people the calculations needed would be too many. Thankfully, we can use a little trick. We want to calculate the probability that two people are born on the same day, which we call p(B), but it’s more simple to do the opposite. So we’re going to compute the probability of two people not sharing their birthday, and we call this p’(B). When we have p’(B), to calculate the probability p(b) all we have to do to get the result is p(B) = 1-p’(B) Let’s start simple. The probability that two people don’t have the same birthday is p’(B) The 365/365 term means that the first person can be born on any day of the year. However, if we want that the second person doesn’t share the birthday with the first one, we have to exclude that day from the number of possible birthdays for the second person. We can do the same for three people and the result is this You probably already guessed where we’re going with this. If we apply this principle for 23 people the result we get is This means that the probability of two people not sharing the birthday is 49.3% if there are 23 people. Now to calculate the probability we simply have to do There we have it, if we take a group of 23 people, it is more likely that two of them share their birthday than not. To better visualize the result it is useful to plot in a graph the two probability we have calculated, p’(B) and p(B). To be more specific, here are the probabilities of two people sharing their birthday:
Conclusion Now you may be wondering why is this problem a paradox. And you would be right because it is not. However, the fact that there's more than a 50% chance that two people are born on the same in a small group of 23 people, is really counter-intuitive. The main reason is that if we are in a group of 23 and we compare our birthday with the others, we think we're making only 22 comparisons. This means that there are only 22 chances of sharing the birthday with someone. However, we don’t make only 22 comparisons. That number is much larger and it is the reason we perceive this problem as a paradox. In fact, the second person was compared with the first one, so he/she has 21 comparisons to make. The third one has to do 20 and so on. To get the total number of comparisons we have to do: So in total, we make 253 comparisons. Those are a lot more than the 22 we thought we were making at the beginning. In conclusion, the counter-intuitive nature of this problem is the reason everyone refers to it as the birthday paradox even though it isn’t. What is the probability that 2 people out of 30 have the same birthday?"there are 30 people, and 365 days, so 30/365 sounds about right. Which is 30/365 = 0.08..., so about 8% maybe?" But no! The probability is much higher.
What is the probability that at least two have the same birthday?This means that any two people have a 364/365, or 99.726027 percent, chance of not matching birthdays.
What is the probability that 2 friends have same birth month?So, required probability =1212×12=112.
What is the probability of two people in a group of 25 having the same birthday?Answer: The Probability that of 25 Randomly Selected Students, at Least Two, Share the same Birthday is 0.5687. Let's solve this step by step.
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