Easy math puzzles

Quote: Ace2

You go to Vegas over a long weekend to play a single-deck blackjack game with a 0% edge ($500 minimum). You decide to play 1,200 hands but you will quit if you bust or double your initial bankroll before reaching 1200 hands.

What size bankroll should you bring to give yourself a 1/3 chance of busting, 1/3 chance of doubling and 1/3 chance of finishing the 1200 hands without busting or doubling?

Assume a standard deviation of 1.1547, flat betting $500 one hand at a time and perfect basic strategy to realize the 0% edge
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Applying my conjecture that the probability of reaching a point at any time during a session is double the probability of ending the session at/beyond that point , we need the z-score corresponding to the probability of 1/3 * 1/2, which is 0.967. Then take 1200^.5 * 1.1547 * 0.967 to get the answer of 38.7 units * $500 = $19,350.

Verification: Knowing the standard deviation and edge, you can easily calculate that this game is statistically equivalent to a bet with a 3/7 probability of winning 7 for 3. Markoving 1200 bets shows that a bankroll of 38 units * $500 = $19,000 gives bust/double/finish probabilities of 33.4%/33.4%/33.2%. I believe this is the closest you can get to 1/3.

So I'd bring $20,000

You don't need that much computing power. You do, however, need to know how to code around the limits of floating-point numbers.

This easy math puzzle is based on the rules of Gin and Win.

Gin and Win
The object of trying to put as many cards as you can in sets, with the goal of minimizing dead wood, is also the goal of Gin and Win. It should be emphasized that pairs and flushes of at least three cards count as sets in Gin and Win. The rules start as follows:

1. Cards are ranked as in poker, except aces are low only.
2. To begin, each player must post an Ante bet.
3. The player and dealer are each dealt seven cards, face down, from a normal 52 card deck.
4. The player looks at his hand and removes any combinations of pairs, trips, and quads (rank melds), or any flushes of three cards or more cards (suit melds). One card cannot be part of more than one set.
5. The remaining cards are known as "Dead Wood" and are set apart from the cards that belong to a set.

Math puzzle:
With what probability will a player be dealt a hand with "no sets" and thus have seven Dead Wood cards?
With what probability will a player be dealt a hand with exactly five Dead Wood cards?