The Fallacy of the Double Winner

The BBC report that a French lottery player has just won the "My Million" lottery for the second time, at odds of 16 TRILLION TO 1 (16,000,000,000,000 : 1). These odds are glibly attributed to unidentified "mathematicians". While it is an interesting story, just how accurate are these odds?

French My Million lottery player scoops second €1m jackpot

I'm going to try and calculate the odds myself and unlike the BBC, I'll show my working out.

First of all though, we need to clarify the question. The story would have been equally valid if it had been any of the previous winners, and if they had won for a second time in any of the subsequent draws. This includes draws that haven't happened yet, the story would simply be published once a player won for the second time. Clearly the chances of this happening improve with each new winner and each new draw. 

So, the question I will try to answer is:

What is the probability of a My Million draw being won by a previous winner?

Some facts about the prize draw apply:

1. The draw occurs twice a week, and at time of writing there have been 585 winners.
2. A random winner is announced at every draw (occasionally more than one).
3. All EuroMillions entrants in France are automatically entered in My Million.

Also we need some assumptions and estimates:

1. Approximately 50 million players take part in EuroMillions at each draw across Europe.
2. Approximately 15% of these players are in France (based on percentage of winners).

So I will estimate that 7.5 million players enter the My Million draw each time.

Based on this, the chance of a second time winner over time is:

Draw #	Previous Winners	Chance this draw	Chance so far
1	0	0	0
2	1	7,500,000 : 1	7,500,000 : 1
3	2	3,750,000 : 1	2,500,000 : 1
4	3	2,500,000 : 1	1,250,000 : 1
5	4	1,875,000 : 1	750,000 : 1
...	...	...	...
586	585	12,820 : 1	43 : 1
...	...	...	...
1001	1000	7,500 : 1	15 : 1

So the true odds of this win are approximately 43 : 1 and as time goes by the likelihood of a double winner just keeps increasing.