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Learning To Count Possibilities

But rather than just considering how to play the hand, let's look at some other things. Such as, if you hold the clubs is it easier to get a straight or a flush?

Let's count the flushes first. Once we are holding the 345, there are 10 more clubs in the deck. So we have any of 10 clubs that can be the first card we draw and once that card is drawn, there are nine clubs that can be the second card we draw. Multiplying those together (9 x 10 =90), we need to divide by two because the order doesn't count. That is, it makes no difference whether we draw the 6c first and the Ac second, or draw the Ac first and the 6c second. That gives us a total of 45 flushes.

These 45 flushes include straight flushes. Since there are three of them (76543, 65432, and 5432A), that leaves 42 regular flushes. There are ALWAYS 45 (flushes + straight flushes + royal flushes) when we draw two cards to a 3-card flush.

There are a few different ways to calculate straights. Here's the way I prefer. Take the 76543 straight first. There are four 7s in the deck and four 6s, and any of the four sevens can be matched with any of the four sixes -- which gives us 16 76543 straights, including the one club straight flush. That leaves us with 15 76543 straights and 15 65432 straights. For the A2345 straight, however, there are only three aces still in the deck because we were dealt the Ah and we threw it away. That leaves 4 x 3 -1 = 11 5432A straights after we subtract out the club straight flush. Therefore we have 15 + 15 + 11 = 41 straights.

So you are slightly more likely to get a flush than a straight starting from holding the three clubs.

Here's another question. If you were drawing three cards to the Ah Jh, would you be more likely to end up with four aces or a royal flush? This is easy. You'd be equally likely as you¹d require three perfect cards in either case. Neither 16,214-to-1 shot would be more or less likely than the other, although getting the royal would be considered "luckier" because it pays more.

Here's another question. With this hand, if the value of flushes were reduced from 6 to 5, would it affect you more if you held the hearts or if you held the clubs?

This one is trickier. We've already decided that there are 42 flushes if you hold the clubs, and perhaps you know that there are 1,081 possible hands you can draw when you hold three cards and draw two. That means about 4 percent of the time drawing two cards to the clubs will produce a club flush (42 / 1,081 = .0389 = 3.89 percent). We haven't gone through how many flushes you'll get if you draw three cards to the suited AJ (I bet you're grateful for that!), but it comes out to 164 times out of 16,215, which equals a little over 1 percent of the time. So pay schedule changes will affect the club draw more than the heart draw.