# Five card draw odds question

You hold:

Ace of hearts
Queen of clubs
3 of clubs
2 of diamonds

Should you discard the ace along with the other two cards in the hopes of bettering your odds to get another queen? Or are your odds of getting either a queen or an ace better if you hold the ace?

I don’t have time to go through the details right now, but I’ve done it before, and you’re better off drawing all three cards.

An intuitive way to think of it is the following. For one thing, it’s unlikely another player has a pair of queens, so your ace kicker probably won’t help much. When you draw the first of three cards, you have a chance of getting another queen. If you don’t get a queen, then you’re back in the position you would have been had you held AQQ–your next two draws could match a queen, or match your kicker.

If nobody else goes through the specific odds (I highly doubt that will happen), I’ll do it later today.

There are 47 cards unaccounted for, of which 3 are Aces and 2 are Queens.
i) Discarding the Ace: the probability that you do not draw a Queen is 45/4744/4643/45, so the probability of getting a Queen is roughly 0.125, or 1 in 8.
ii) Keeping the Ace: the probability that you do not get either a Queen or an Ace is 42/47*41/46, so the probability that you do is 0.204, or slightly better than 1 in 5.

One other point: I have only calculated the probabilities that you improve your hand. In a game you would also need to consider the likelihood of each hand winning the pot.

So, you’re getting into the poker thing, then?

Thanks, Cabbage!

Oops, didn’t see your post, Jabba. Thank you, too! Edlyn got a Hoyle Casino game, and I’m dabbling in it. Your advice seems to contradict Cabbage’s. Any idea why he advised drawing 3?

I would guess it is connected with the point about the value of the hands.For example, if you draw 3 then you have a 1 in 8 chance of ending up with 3 Queens or better, a good hand. Dawing 2, you have a 1 in 5 chance of improving your hand but this includes the possibility of having 2 pairs, not as good. If you draw only 2 your chances of getting 3 Queens are worse. I’ll play with the numbers a bit more and get back to you.

Okay, back-of-an-envelope calculations give me the following:

(i)If you draw 3 cards
4 Queens, probability 0.003
Full House, 3 Queens and 2 of something else, probability 0.008
Full House, 3 of something and 2 Queens, probability 0.002
[Thus the probability of a Full House is 0.010]
3 Queens, probability 0.114
2 Pairs, probability 0.160

(ii) If you draw 2 cards
4 Queens, probability 0.00006
3 Queens, 2 Aces, probability 0.006
3 Aces, 2 Queens, probability 0.003
[Thus the probability ofa full house is 0.009]
3 Queens, probability 0.078
2 Pairs, probability 0.173

Thus the only probability that is increased by drawing 2 cards is that of getting 2 pairs.
Draw 3.

Wow. Thanks!

Lib: Against the computer, draw 3, for the reason already given (i.e. you’re more likely to improve your hand that way). Against other people, however, it would sometimes make sense to hold on to the Ace. You’re only slightly less likely to improve your hand (though you’re considerably less likely to get a third queen), and it makes your hand harder to read (you can represent with your betting three of a kind, for example). Depends on the circumstances and your proferred style of play.