What is the probability of having a length of gestation less

What is the probability of having a length of gestation less than or equal to 36 weeks given that the infant is low birth weight? From Rosner, Fundamentals of Biostatistics pg 62 #3.50

Solution

I am assuming that you need answer of following question:

Question

Infants are classified according to low/normal birth weights and length of gestation. Assume the probabilities of periods of gestation as given in the table below. Also, assume that the probability of having a low birth weight given that the length of gestation is <20 weeks is 0.540, the probability of having a low birth weight given that the length of gestation is 20-27 weeks is 0.813, the probability of having a low birth weight given that the length of gestation is 28-36 weeks is 0.379 and the probability of having a low birth weight given that the length of gestation is >36 weeks is 0.035.

Length of gestation Probability
<20 0.0004
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20-27 0.0059
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28-36 0.0855
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>36 0.9080

Let L shows the event that infant is low birth weight. So we have following probabilites:

P(L|<20 weeks)=0.540

P(L|20-27 weeks)=0.813

P(L|28-36 weeks)=0.379

P(L|>36 weeks)=0.035

So required probability P(L) will be:

P(L)=P(L|<20 weeks)*P(<20 weeks)+P(L|20-27 weeks)*P(20-27 weeks)+P(L|28-36 weeks)*P(28-36 weeks)+P(L|>36 weeks)*P(>36 weeks)

=0.540*0.0004+0.813*0.0059+0.379*0.0855+0.035*0.9080=0.000216+0.0047967+0.0324045+0.03178=0.0692

Hence, required probability is 0.0692.