Not sure about your poker odds calculator, seems to me your input is correct, but the result is not. As you said yourself, we all know that a pp vs two higher cards is about 50/50. The vast majority of this 50% chance for the two high cards comes from making a pair that beats the pocket pair.
Just checked and there is a bug in that online calculator you linked to. If you enter 50 cards, 5 drawn cards, 50 outs, and 1 out needed it gives you... zero probability! Should obviously be 100%. With the same parameters, if I vary the number of outs, it starts decreasing your probability to draw at 11 outs (less chance to draw with 11 outs than with 10!!). I wrote to the guy to tell him.
You cannot use what you call the direct method. Here is a simple example: you bet on tails and you flip a coin twice. If tails comes at least one time, you win. What are you odds of winning? According to you, it would be 1/2 + 1/2 = 1 -> 100% chance to win. This is of course not the case because the coin could fall on heads twice. So the proper way is to calculate how often you loose: how often does the coin fall on heads twice? 1/2 * 1/2 = 0.25 -> you have 25% chance of loosing -> you have 75% chance of winning.
Similarly, with your method, if there were 9 board cards instead of 5 in your example above, we would have 9/8 chances to win (112.5%)!?
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