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If you know nothing about the parents' risk for these diseases or other contributing factors (race, environment, etc.), then the answer depends solely on the probabilities of getting any one of those diseases.
Let's say the risks are P1, P2 and P3 where P1 is the probability of the baby having rare disease #1 and so on. A probability of 1 is a sure thing and 0 means no chance, so the probability of a rare disease is a number slightly greater than 0.
The risk of getting disease #1 AND #2 AND #3 is the product: P1xP2xP3.
If, for example, the each risk is one in a million (P1=P2=P3=0.000001), then the odds of having a baby with all three is 1 in 1,000,000,000,000,000,000.
The math changes severely if one of the parents is at risk for the disease. Let's say the mother has one gene that causes the disease #1. Then P1 is no longer 1 in a million, it is 0.5. If the parent's racial group is afflicted with P2=0.001 as well, then P=0.5x0.001x0.000001=0.000000005, which is 500,000 the risk of others.
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