the trick of it

Suppose we have 100 mortgages that pay $1 or $0. The probability of default is 0.05. We pool the mortgages and then prioritize them into tranches such that tranche 1 pays out $1 if no mortgage defaults and $0 otherwise, tranche 2 pays out $1 if 1 or fewer mortgages defaults, $0 otherwise. Tranche 10 then pays out $1 if 9 or fewer mortgages default and $0 otherwise. Tranche 10 has a probability of defaulting of 2.82 percent. A fortiori tranches 11 and higher all have lower probabilities...