In a genetic study, an allele $A$ occurs in $10\%$ of the population. A test for the presence of the allele has a false positive rate of $5\%$ and a false negative rate of $10\%$. If an individual tests positive, the probability they actually carry the allele is $P_1$. If the same individual is tested a second time independently and tests positive again, what is the new probability $P_2$ that they carry the allele?