Birthday paradox: meaning of random

Question

In the wikipedia page (http://en.wikipedia.org/wiki/Birthday_problem) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some pair of them will have the same birthday. We assume that that each day of the year is equally probable for a birthday."

My question is what is the meaning of "randomly chosen" here ? Is the assumption of equally probable for a birthday needed separately ? Does not the word "randomly chosen" imply the equal probability ?

Answer 1

The assumption is that each person's birthday is chosen randomly from the $365$ days of the year.

Answer 2

Random, in its every-day meaning, simply means that uncertainty is involved. In the precise mathematical meaning it means that uncertainty is potentially involved. In any case, random does not imply any uniformity over the possible outcomes. Tossing a die with six sides numbered 1,2,3,4,5,6 produces a random event with equal probabilities for each outcome. Tossing a die with six sides numbers 1,1,1,1,1,6 produces a random event with possible outcomes 1 and 6, with unequal probabilities. Purely mathematically, tossing a die with six sides numbered 6,6,6,6,6,6 produces a random event with one outcome $6$, whose probability of occurring is $1$.

The assumption on birthdays is that each day of the year is as likely as any other, thus that the distribution is uniform.

It should be noted that the assumption of uniformity in face of uncertainty is a very common mistake. Not only it is wrong in many finite situations, in certain infinite situations it can easily be shown that no uniform distribution exists (this leads to some paradoxes, like one variant of the two-envelope paradox).

Answer 3

As in the comment from Alex, the problem assumes a discrete uniform distribution. The word uniform means that each possible outcome is equally likely.

To help clarify the distinction with the word random, consider the following random variable. You flip a quarter, dime, and nickel and record the total number of heads as the variable $H$. The possible values of $H$ are $\{0, 1, 2, 3\}$. The results of flipping the coins is certainly random and the distribution is discrete.

However, the different outcomes are not equally likely. In fact the probabilities are as follows. (This is an example of the binomial distribution.) $$ \begin{array}{c*{3}{|c}} h & 0 & 1 & 2 & 3 \\ \hline P(H=h) & \tfrac{1}{8} & \tfrac{3}{8} & \tfrac{3}{8} & \tfrac{1}{8} \end{array} $$

Answer 4

"Randomly chosen" concerns independence here. Later in the same article,

"When events are independent of each other, the probability of all of the events occurring is equal to a product of the probabilities of each of the events occurring."

That is the rationale for the "randomly chosen" part.