The Birthday Paradox can be restated as inquiring into the probability that if a number of people participate in a sequence of lottery drawings, someone or other of these will win the lottery more than once.We shall assume that each participant in the lottery has a distinct number and that exactly one or these numbers will be drawn to determine the winner.
Let be n the number of participants in the drawings and d the number of drawings. Let PERM(n,d) represent the number of permutations of n different things taken d at a time. The number of ways the d winners could be all different people is PERM(n,d) and the number of ways d winners could be anyone at all is nd, so that the probability that there are no duplicate winners is
q = PERM(n,d)/nd. The probability of at least one multiple winner, therefore is p = 1 - q.Working out a few examples leads to an unexpectedly high probability of a multiple winner. For example, if a lottery has 10,000 participants and there is a drawing every day, there is a probability of 0.8093 of a multiple winner within six months and .9988 of a multiple winner within in a year. This probability is so much higher than the probability that any particular person will be a multiple winner that whoever this multiple winner is may very well come to the conclusion that the result is not due to chance.
If he has used some sort of mathematical system to pick lottery numbers he may take his luck as proof that his system is mathematically valid and try to sell it to the world. You will see an example of that here.