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You have 23 chromosome pairs, and in each one of those pairs, there's one chromosome from your mom and one from your dad. Assuming that when a gamete forms, you get one chromosome from each pair, and the chromosomes are assigned randomly to a gamete, then there are 2^23 possible combinations, or about 8.4 million. A gamete with the exact same chromosomes you got from one of your parents are two of those possibilities, so the probability is 2 in 8.4 million, or 1 in 4.2 million.
Think of a series of coin flips, where heads indicates you take your dad's chromosome, and tails indicates you take your mom's. For each flip, the probability of getting heads is 1/2. Since the flips are independent events, probabilities multiply. Getting two heads in a row has a probability of 1/2 * 1/2, or 1 in 2^2. Getting 23 heads in a row (creating your dad's gamete) is where you get the 1 in 2^23, or 1 in 8.4 million odds.
This brings up an interesting question - we've just figured out the odds that you could produce a gamete identical to your parents', and that's identical to the odds of producing any one particular gamete, such as the ones your dad and your mom produced when you were born. So, what are the odds that, if your dad and mom had another child, that it would be genetically identical to you?
Well, your dad has to contribute the same gamete, so that's 1 in 2^23. And your mom has to contribute the same gamete, so that's 1 in 2^23 as well. They're independent events, so the probabilities multiply, so you get 1 in 2^23 * 2^23 = 2^46 = 70,000,000,000,000, or 70 trillion!
I hope that helps!
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