## Thursday, October 22, 2009

### Good Hand

Pretend you have five cards: a ten, a jack, a queen, a king and an ace. Shuffle the cards together, then place the cards facedown. Select four cards, returning each card and re-shuffling the deck after each pick, what kind of hand would you more likely draw: four aces or a straight picked in sequence?

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Math

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I'd say the straight.

ReplyDeleteFor the 4 aces, you have exactly one card of five to pick each time. So I think the chances would be:

(1/5) * (1/5) * (1/5) * (1/5)

For the straight, you can start with a ten or jack, then you have to pick exactly the last three:

(2/5) * (1/5) * (1/5) * (1/5)

Also, the rules don't really say the straight can't go backwards, so you could start with the ace or king and go backwards too, so that should be:

(4/5) * (1/5) * (1/5) * (1/5)

So no matter what, I'd say the straight is more likely, but it's been a while since I did probability in school, so feel free to point out where I'm wrong!

The straight.

ReplyDeleteThere's only one combination for the Aces but there are two combinations for the straight.

The trouble with getting four aces is the trouble with picking the first ace. If you're going for the straight, it really doesn't matter what the first card is. That's assuming you can start the straight from any card: for example, Q, K, A, ten.

ReplyDeleteIf you don't allow the straight to go around like that, then the only card you can't pick up first is the Q.

Andy did a great job laying out the odds for both, so I'll leave it at that.

So this isn't a normal deck, then?

ReplyDeleteNormal? There are only five cards.

ReplyDeleteI'm not sure I understand the question Jered?