Math Made Easy: Problem of the Day 8
For today, a quick probability problem. Probability is an interesting topic to me, because the entire world works off it. The universe is nothing but probabilities – there is no such thing as 100% until an event has happened, until it is in the past. Nowhere is this more true than in the various games of chance we humans like to play. Personally, I don’t play most of them myself – I don’t gamble. And what I mean by that, is I won’t let the chances of my winning something be left up in the air and vague. Before I do something, I want to have calculated the probabilities involved, and I want them stacked as heavily in my favor as possible. And that’s how I play Texas Hold ‘Em.
For those of you not familiar with the game, here’s how it goes. It’s a form of Poker, where each player is dealt two cards that only they get to see. An initial round of betting is made based on those two cards, and when it’s complete, the dealer lays three cards face up for all to see – this is called the Flop. Players are allowed to build their hands with the two cards they hold plus any of the face up cards on the table. After another round of bets, the dealer places one more card face up on the table (the Turn). Players get to reassess their hand with that, and after a final round of betting, the dealer places a final card face up on the table (the River), resulting in one last round of betting before all cards are revealed and hands compared (if one player didn’t get everyone else to fold first). The order of which hand beats what is the same as any regular Poker game.
So, for our problem, you’re playing Texas Hold ‘Em, with a single deck. It’s just you and another guy. You’ve got a four of hearts and an seven of hearts in your hand. The table is showing a 3 of clubs, queen of hearts, king of spades, and an ace of hearts. Just the river is left.
The question is, should you make a call? Should you fold? What are the actual odds of you getting that flush?
Let’s examine what possible hands you have, and what could your opponent have, not knowing what two cards they’re holding. With a 4 – 7 pocket, you’re probably not going to get high card, and you haven’t paired anything. The only thing you could be in on is a flush – you just need one heart on the river. Your opponent COULD have 3 of a kind already, if they’ve got a pocket pair. Which means they could also be in on a four of a kind. And there’s a potential straight on the table – your opponent could have one already if they have a ten and a jack of any suit, or if they have either of those and the other comes on the river. Oh, and don’t forget, they could be in on that flush, and since you don’t have high card, their flush will probably beat yours on high card if they get that.
What beats what? If you get the flush, you beat any pairs, the three of a kind, and any straight. Assuming you get the flush, your opponent could only beat you on a 4 of a kind. So let’s examine the odds of both of those, and discount the odds of the other hands. We don’t need to consider them because you lose if you don’t get the flush anyway, and, if you do, you win unless your opponent gets that four of a kind.
Really, we only need to calculate the odds of your flush. One might think they’re a straight 1 in 4 since there are four suits, each with an equal number of cards. But that’s not the case anymore since not all cards are in the deck at this point. Currently, 8 cards have been dealt, so there are only 44 total cards in the deck. Of those no longer in the deck, at least 4 are hearts. Since this is what you need to win, let’s assume your opponent also has two hearts in their hand to make the probability of your win as low as possible. That means 6 hearts out, only 7 remaining in the deck. So the odds of you getting that flush are only 7 in 44 (not one in 4 – seven possible hearts in the deck out of 44 remaining cards), which means you only have a 16% chance of getting a flush. Just for giggles, your BEST chance of getting a flush at this point – that is, if your opponent *doesn’t* have any hearts – is 9 in 44, which is still only 20%. Note that I simplified this – getting the exact odds of your flush is possible, but beyond the scope of this article. (It turns out to be 19.55%)
If it were me, at this point, seeing that you’re beaten by almost any pair, you don’t have a chance at high card, and there’s a chance your opponent is in on the flush (actually the same as yours since they could also have two hearts) and the straight as well as the possible 3 or 4 of a kind – I’d fold unless my opponent is betting conservatively, or I think I have a chance of bluffing him (but then, I would have been setting that up from the start).
Now, it is possible to calculate the exact win chance you have (only about 1 in 3, actually, even more reason to fold), but I’m not going to get into that for this article – it’s more complex (it starts involving double factorials and dealing with permutations in the order of 10^{6}), and not something most people would be able to do in their heads at the poker table without their opponent catching on to what they’re doing, and in a casino, whipping out paper or calculator to figure the odds would get you kicked off the table.
