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How can I calculate the odds of something happening twice?

Example:

I know my odds of losing with AA vs AK are just 7%, but what are they to lose with AA vs AK both times I have AA in a row?

How do I take the odds of it happening once, and figure out the odds of it happening the next time I have it?

Am I right in assuming that if I had AA 20 hands ago, and I lost to AK, the odds of it happening again the very next time I have AA are far smaller?

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Am I right in assuming that if I had AA 20 hands ago, and I lost to AK, the odds of it happening again the very next time I have AA are far smaller?

Yes and no. But mostly no.

The odds of AA losing to AK never change, as long as all of the variables are the same. What happened in the past is in the past, and has no bearing on what will happen this time or next time. If it is indeed 7%, then it is still 7% even when you lost last time.

Now that being said, if you want to determine the odds of you losing in that situation the next two times that it happens then yes, now the odds of that happening are much lower because both events have the possibility of winning (whereas in the first example, the first event already lost so you only have to worry about one event). You calculate that by multiplying the percentage chance of each event happening by each other. So, in this case, 7% * 7% = 0.49%, or about 1/2 of 1 percent.

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