Birthday Paradox is true. The probability of at least two of 23 people sharing a birthday is, surprisingly to most people, a little better than 50/50. The probability that all 23 people have different birthdays is:
(364/365) * (363/365) * (362/365) * ... * (342/365)
which comes to:
364*363*362*361*...*342
365^23
which is slightly less than 0.5. So the probability that at least two share is the complement of this, or just above 0.5.
Monty Hall problem: It is indeed better to switch. Whether the door you selected has a car or a goat behind it, the host is ALWAYS able to open one door you didn't pick that has a goat. The two keys are: (1) the host knows which door has the car, and (2) the host will ALWAYS open a door with a goat behind it. Since the probability that you selected the door with the car is 1/3, and now we KNOW that of the two doors left, one has a car and the other has a goat, there is a 2/3 chance that switching will get you the car.
Boy/Girl problem: It's been proven in the sense that, if Mr. Smith has two children, it matters how you found out that at least one child is a boy. If all he told you was they were not both girls, that eliminates one of (GG, GB, BG, BB) from the set of possibilities, leaving (GB, BG, BB) and a probability of 1/3. If you happened to see one particular child and it was a boy, that effectively eliminates GB from the list too if we figure the first one is the child you saw. That makes the probability 50/50.
20 Sep 2011, 10:33 am
