What is the probability?

Question

There are 30 students in a classroom. What is the probability 2 of the 30 students will have the same birthday?

I need an explanation and whether it is a dependent event, independent event, inclusive event, or a mutual exclusive event.

Answer 1

The simple method is to
1) work with a 365 day year
2) find the prob that they all have unique birthdays, then subtract this from 1.

P(unique birthdays) = (365/365)*(364/365)*(363/365) ...*(336/365)
= (365P30) / 365^30
= 0.2937

P(common birthday) = 0.7063

Regarding the last part of your question, dependence and exclusivity do not apply to single events. You can't say event A is independent or exclusive. You have to say A and B are independent or mutually exclusive.

Answer 2

This is a classic probability problem and the answer seems unreasonable but it turns out to be about 70%%. To calculate, it is easier to find the probability that everyone in the class has a different birthday:
(365*364*363*362*......*337*336)/(365^30)=.29368

Subtract from 1 to get the probability of 2 people in the class having the same birthday.

1-.29368=.7063 or 70.63%

Two people having the same birthday would be an independent event, meaning that 1 person's birthday doesn't depend and isn't affected by anyone else's.

Answer 3

It's usually easier to figure the probability that nobody has the same birthday and then subtract from 1.

For the sake of simplifying this I will assume 365 days in a year, but technically we would have to account for leap year birthdays.

The probability that the 2nd person has a different birthday than the first is:
364/365

The probability that the 3rd person has a different birthday than the first two is:
363/365

The probability that the 4th person doesn't match any of the 3 prior birthdays is:
362/365

etc.

In general, the probability that the nth person doesn't match the prior (n-1) people is:
[365 - (n-1) ] / 365

So the chance that none of them match is:
364 . 363 . 362 . etc... 337 . 336
----- x ----- x ----- x ..... x ----- x -----
365 . 365 . 365 . ...... . 365 . 365

Figure it out and subtract from 1 and you'll have the probability. It's around 70% and gets bigger as you add more people. I believe 23 people is the point where it tips from 50% to more likely.

Answer 4

It's independent; The birthdays of kids in the class are independent of the fact that they're in there (unless, of course, they took all the May birthdays and put them in the same class, etc...)

Think about p(two people have the same birthday): First person has a birthday, the other person has a 364/365 chance of not being born on that same day. The next person would have a 363/365 chance of not being born on the same day as either of the first two.

So the chance of 30 people not having the same birthday is (364*363*362*...*335)/365^30.

Answer 5

fifty percent chance

Answer 6

.