what is the probability that all four family members are assigned to the same station?
Probability of Shared Birthdays
or, How to Win Money in Bar Bets
Copyright � 2001�2022 past Stan Chocolate-brown, BrownMath.com
Summary: In a grouping of 30 people, would you be surprised if ii of them have the same birthday? As it turns out, you should be more surprised if they don�t.
There are 365 possible birthdays. (To keep the numbers simpler, we�ll ignore bound years.) The key to assigning the probability is to recall in terms of complements: �Two (or more) people share a altogether� is the complement of �All people in the group have dissimilar birthdays.� Each probability is i minus the other.
(a) What is the probability that whatsoever 2 people have unlike birthdays? The first person could take any birthday (p = 365�365 = 1), and the 2d person could then take any of the other 364 birthdays (p = 364�365). Multiply those 2 and you have almost 0.9973 as the probability that any two people have different birthdays, or 1−0.9973 = 0.0027 as the probability that they have the same birthday.
(b) At present add a third person. What is the probability that her birthday is different from the other two? Since there are 363 days still �unused� out of 365, we have p = 363�365 = virtually 0.9945. Multiply that past the 0.9973 for 2 people and you have about 0.9918, the probability that three randomly selected people will take different birthdays.
(c) Now add a fourth person, and a fifth, and then on until you have 22 people with different birthdays (p ≈ 52.4%). When you add the 23rd person, you should have p ≈ 49.3%.
(d) If the probability that 23 randomly selected people have dissimilar birthdays is 49.3%, what is the probability that 2 or more of them accept the aforementioned altogether? 1−0.493 = 0.507 or l.7%. In a randomly selected grouping of 23 people, information technology is slightly more probable than not that two or more of them share a birthday.
For northward people (n ≤ 365), your chain of n fractions would exist
and therefore
On your TI-83, to get 365P n you first enter the 365, then press [MATH] [◄] to become the PRB menu and [2] for nPr, then enter the second number (n). In Excel, it�s PERMUT(365,n).
What if n > 365? In this case there is no need for any calculations (and in fact the above formula won�t work). If there are 366 or more than people, merely just 365 possible birthdays disregarding leap yr, so two or more of them must share a birthday.
Hither are some sample results:
| Probability in a grouping of n people that 2 or more accept the same birthday | |
|---|---|
| 10 | 0.117 |
| 20 | 0.411 |
| 22 | 0.476 |
| 23 | 0.507 |
| thirty | 0.706 |
| 40 | 0.891 |
| l | 0.970 |
Yous can run across that the dividing line is betwixt 22 and 23 people. In a group of 22 people, the odds are less than 50�50 that ii share a altogether; in a group of 23, the odds are meliorate than 50�50. In a bar with even a pocket-sized oversupply, if you can get someone to take your bet that two people share a birthday, you�ll win more ofttimes than yous lose.
In a randomly selected group of 50 people or more than, information technology is near certain that two or more volition share a birthday (p ≥ 97%). On a crowded Fri nighttime you can really clean up � if nobody else in the bar knows probability!
Excel Workbook
Y'all can download an Excel workbook to compute the probability of shared birthdays for whatever size group.
The workbook uses two methods to compute these probabilities. Method one uses the formula shown to a higher place. You�ll notice #NUM errors for groups larger than northward = 120, because 365121 is bigger than the largest number Excel can handle. In this particular case that�south non a problem, because the probability is finer 1 for n > 118 anyway, simply even so all those #NUM cells look foreign.
Method 2 solves this past computing the probability for each size group from the probability for a group one person smaller, just as I did in steps (a) through (d) to a higher place. That way Excel never has to deal with numbers bigger than 365 in whatsoever one stride, and the #NUM errors are avoided.
What�s New?
- 19 Sept 2020: Added an Excel workbook.
- 25 May 2003: New article.
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Source: https://brownmath.com/stat/birthday.htm
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