Act Of Defiance

Snaggle Board is a betting game which I believe was invented by Adrian Dracup. It works like this:

1. All players make a bet
2. The dealer shuffles a 52 card deck
3. The dealer deals out all the cards in a line, one at a time

If any consecutive 5 cards in the deck make a straight or better poker hand, then the dealer pays out an amount to each player equal to double the amount they staked. Otherwise the dealer takes all the money.

What are the odds of the players winning in this game?

The odds against getting a straight or better in any random 5 card hand are 131:1 (or 1/132)

The snaggle board constitutes 48 random 5 card hands (you could choose to examine each set of 5 consecutive cards in any order, so the odds of any of them being a straight or better is not effected by the order you examine them in). So answering the question is equivalent to the finding of the 1 – the probably that all 48 hands are worse than a straight

p(players wins) = 1 - p(all 48 hands worse than a straight)

The probability that a hand is worse than a straight is

p(hand worse than straight) = 1 - 1/132 = 131/132

The probability that a sequence of 48 hands are all worse that a straight is

p(48 hands worse than a straight) = (131/132)^48
p(48 hands worse than a straight) = 0.694

Therefore the probability that the players will win is

p(players wins) = 1 - p(48 hands worse than a straight)
p(players wins) = 1 - 0.694 = 0.306

So there is approximately a 30% chance that the players will win, and a 70% chance that the dealer will win.

Easy money!

correction: the dealer pays out double the stake, not equal to the stake as previously stated