Math and Bias

You are offered a bet where a board of 20 cards will be laid out from a 45-card deck. 7 black cards have been removed from a standard 52-card deck. You can pick one of the following sequences of suits. If your pick shows up in the laid-out cards, you win $100. You have to pick immediately, without running through the math. Which do you pick? (d=diamonds, s=spades, c=clubs. h=hearts)

1.- dschcs
2.- scscdcsc
3.- hdschcsd

or simplified (r=red, b=black):

1.- rbbrbb
2.- bbbbrbbb
3.- rrbbrbbr


Now we're all smart people here, right? We know some probablity. #2 is right out because of all the blacks, and only 1 red. So it follows that #3 is the smart pick, since its 50/50 split is the closest statistically to the cards in the deck.

Right?

Wrong.

Look at #1 again... and compare it to #3. #1 is a subset of #3, meaning that for #3 to happen, #1 HAS to happen, PLUS you need an additional heart at the beginning and diamond at the end. P(A & B)<=P(A). It's called a conjunctive fallacy and is a fairly common bias when asked what seems most likely without doing the math behind it. It's also why lotteries that have 7 numbers pay more than ones with 6, or why more numbers matched on a Keno card pay out more money.

So how does this relate to poker? You'll learn more if you think about it yourself. But I'll give an instance: one word, twice - runner-runner.