MGMT DISC 2

3

Glory.

The Birthday Problem 

P (2 people have different birthdays) = 364 / 365

P (3 people have different birthdays) = (364/365) * (363/365)

P (4 people have different birthdays) = (364/365) * (363/365) * (363/365)

probability that at least 2 of the people in the class share the same birthday = 1 - probability that none of the people share same birthday

= 1 - (364/365) * (363/365) * (363/365) * ........ * (343/365)

= 1 - 365P23 / 36523

= 1 - [ 365! / (365 - 23)! ] / 36523

= 1 - 0.4927

0.4927 is the probability that out of 23 people, no two have the same birthday.

= 0.5073 [Answer]

 0.5073, which is the chances at least two people out of 23 have the same birthday.

As part of probability theory, the birthday problem examines if there is any chance that at least two random persons will have the same birthday. In a group of 23 persons, the probability of having a birthday that falls on the same day is more than 50%.

Ashlynn.

The Birthday Problem

23 x 22/ 2 = 253

1-(1/365)=364/365

(364/365)^253 = .4995

1-.4995= .5005 or 50.05%

or a 50% chance two people will have the same birthday. 

There are 23 people in total, and while you would want to calculate the amount of a single person having 22 pairs. The calculation takes into account everyone's pairs. So, 23 people multiplied by 22 pairs and divided by the 2, the amount of people we want to find that have the same bday. Now we calculate the frequency of 365 days in a year. Out of 365 days in the year, the chance of the birthday falling on the same day is 1/365. The chance on different days is 364/365. We take that fraction and multiply it by the power of 253, the number of pairs. This comes out to .4995. Or a 49% chance percent chance that all 253 comparisons contain no matches. The odds that there is a birthday match in those 253 comparisons is 1 – 49.95% = 50.05%

Another way to think about it is the equivalent of flipping a weighted coin, where there is a 99% chance it will come up heads and a 1% chance it will come up tails. After 253 flips, there's a 50% chance tails would have come up. 

Reference:

Better Explained. (n.d.). Understanding the birthday paradox – BetterExplained.  https://betterexplained.com/articles/understanding-the-birthday-paradox/#:%7E:text=23%20people.,at%20least%20two%20people%20matching.