Probability of Getting No Hand
You can bet on anything. You can even bet on the worst hand possible, 2,3,4,5,7 of mixed suits. (The odds of being dealt this particular combination are 1/2548 by the way). To calculate the high card hand we can simply take all of the hands we have dealt with already and then subtract them from the total possible hands. Below is a list of the total possible combinations for each type of hand:
Straight Flush

40

4 of a Kind

624

Full House

3744

Flush

5108

Straight

10,200

3 of a Kind

54,912

Two Pairs

123,552

One Pair

1,098,240

Total

1,296,420

Now we must subtract this total from the number of possible hands:
2,598,960  1,296,420 = 1,302,540
Now it is a simple matter of dividing our high card hands by the total possible hands:
1,302,540/2,598,960 = 1 in 2
This gives us about a 50% chance of getting nothing. But this also tells us that we have an even chance of getting at least a pair.
Improving a high card hand can be tricky. The best strategy is probably to keep your two highest and hope for a pair or better. Alternatively keep whatever cards you have of the same suit and hope for a flush. Any close runs are also a possibility, however, going for the inside straight is questionable if the raises are high.
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