IBM Ponder This #045 · January 2002

This month's puzzle comes from James Shearer.

To celebrate the new year, you have decided to open a gambling casino.
Because you are a fair-minded person, you will arrange the games
so that your advantage is minimal.
In each game, the gambler will bet one euro coin;
With probability 0.499 the gambler will win (and you, the "house",
will pay him one coin);
With probability 0.501 the gambler will lose (and the "house"
will collect one coin from him).
The outcome of each game is independent of previous games.

You start your casino with k coins in the treasury.

You may have a run of bad luck; with probability 0.499**k,
the first k games will go against you, and the house will go broke.
(That is, its treasury will go to 0 euros.)
Or you may lose k+3 out of the first k+6 games, and still go broke.
And so on.
Part 1:
What is the minimum integer k such that with
probability exceeding 1/2, the house will never go broke?

Part 2:
Use the value of k from part 1.
One bet is placed per hour, starting at the stroke of midnight.
Assuming that the house does go broke, when is this most likely to
happen? (Date and hour.)

Part 3 (off the wall and unrelated):
My cat died last month at the age of seventeen.
We used to enjoy her standing on her hind legs to beg for cream cheese.
Why do we think of her as being futuristic?

Solution

Best opened after a real attempt

To be added.