What is the least number of people that need to be in a room such that there is greater than a 50% chance that at least two of the people have the same birthday?
Only 23 people need to be in the room.
Our first observation in solving this problem is the following:
(the probability that at least 2 people have the same birthday + the probability that nobody has the same birthday) = 1.0
What this means is that there is a 100% chance that EITHER everybody in the room has a different birthday, OR at least two people in the room have the same birthday (and these probabilities don't add up to more than 1.0 because they cover mutually exclusive situations).
With some simple re-arranging of the formula, we get:
the probability that at least 2 people have the same birthday = (1.0 - the probability that nobody has the same birthday)
So now if we can find the probability that nobody in the room has the same birthday, we just subtract this value from 1.0 and we'll have our answer.
The probability that nobody in the room has the same birthday is fairly straightforward to calculate. We can think of this as a "selection without replacement" problem, where each person "selects" a birthday at random, and we then have to figure out the probability that no two people select the same birthday. The first selection has a 365/365 chance of being different than the other birthdays (since none have been selected yet). The next selection has a 364/365 chance of being different than the 1 birthday that has been selected so far. The next selection has a 363/365 chance of being different than the 2 birthdays that have been selected so far.
These probabilities are multiplied together since each is conditional on the previous. So for example, the probability that nobody in a room of 3 people have the same birthday is (365/365 * 364/365 * 363/365) =~ 0.9918
More generally, if there are n people in a room, then the probability that nobody has the same birthday is (365/365 * 364/365 * ... * (365-n+2)/365 * (365-n+1)/365)
We can plug in values for n. For n=22, we get that the probability that nobody has the same birthday is 0.524, and thus the probabilty that at least two people have the same birthday is (1.0 - 0.524) = 0.476 = 47.6%.
Then for n=23, we get that the probability that nobody has the same birthday is 0.493, and thus the probabilty that at least two people have the same birthday is 1.0 - 0.493) = 0.507 = 50.7%. Thus, once we get to 23 people we have reached the 50% threshold.
A man is sitting in a pub feeling rather poor. He sees the man next to him pull a wad of £50 notes out of his wallet.
He turns to the rich man and says to him, 'I have an amazing talent; I know almost every song that has ever existed.'
The rich man laughs.
The poor man says, 'I am willing to bet you all the money you have in your wallet that I can sing a genuine song with a lady's name of your choice in it.'
The rich man laughs again and says, 'OK, how about my daughter's name, Joanna Armstrong-Miller?'
The rich man goes home poor. The poor man goes home rich.
What song did he sing?
A man was found dead with a cassette recorder in one hand and a gun in the other.
When the police came in, they immediately pressed the play button on the cassette.
He said "I have nothing else to live for. I can't go on," then the sound of a gunshot.
After listening to the cassette tape, the police knew that it was not a suicide, but a homicide.
How did they know?
If the man shot himself while he was recording, how did he rewind the cassette tape?
The front of me is the source of a song
Or to kiss with a fervor of love lifelong.
My back is a plant fit for a queen,
Crafted by needle, chemical, or machine.
If measured from the center of the planet, which mountain on Earth is the highest? (it´s not the Everest)
Mount Chimborazo, in Educator. Everest is the highest above level sea level, but because of the way the Earth bulges at the center, Chimborazo (6010 meters, or 20 600 feet above sea level) is the highest when measured from the center of the planet. To put it another way, it is the furthest out into space.
Imagine John, a party magician, is carrying three pieces of gold each piece weighing one kilogram. While taking a walk he comes to a bridge which has a sign posted saying the bridge could hold only a maximum of 80 kilograms. John weighs 78 kilograms and the gold weighs three kilograms. John reads the sign and still safely crossed the bridge with all the gold. How did he manage this?
John is a juggler. When he came to the bridge he juggled the gold, always keeping one piece in the air.
