A while back, @Dan_Eastwood posted this slide:
Across the top are the different markers (e.g CSF1PO, TH01) which are short tandem repeats (STR’s). On the left hand side are the number of repeats in that marker which is defined as an allele. Below each header for each genetic marker is the frequency of the allele in the human population. Using CSF1PO as our example, 5.3% of the population has a CSF1PO allele with 7 repeats, 6.0% have an allele with 8 repeats, and the most common allele is one with 12 repeats found in 29.8% of the population. The slide appears to be cropped, but you can see that there are multiple markers each with multiple alleles.
So let’s do the math. Here is an example of what a DNA profile might look like, keeping in mind that the human genome is diploid so we have two alleles for each gene. I will list the gene and the number of repeats in the allele.
CSF1PO: 9, 12
TH01: 6, 8
TPOX: 7, 10
D5S818: 10, 13
D7S820: 9, 11
Let’s say another DNA sample shared the first allele in each pair (9, 6, 7, 10, 9). What would the chances that two random people would share those 5 alleles? All you have to do is multiple the allele frequencies together. This would be:
(0.037 * 0.124 * 0.017 * 0.070 * 0.109) = 5.9510948e-7 or 1 in 1,680,363.
[note: I am pretty sure I have the math wrong in a fundamental way since it probably requires using nCr or a comparable formula, but it is going to be similar to what I have above.
If someone has the time and knowledge perhaps they could be kind enough to fix my math for me.]
Obviously, the chances of two people sharing lots of markers due to random chance is really, really low. The better explanation for sharing that many markers is common ancestry.
[edit: the last paragraph was a mess]