Question 1113309: Consider the following natality statistics for a given population. According to these data the probabilities that a randomly selected woman who gave birth was in each of the follwing age groups is given in the table.
<15years=0.003
15-19years=0.124
20-24years=0.263
25-29years=0.29
30-34years=0.22
35-39years=0.085
40-44years=0.014
45-49years=0.001
a)Given that the mother of a particular child was under 30years of age , what
is the probability she was not yet 20years?
b)Given that the mother was 35 years of age or older, what is the probability
she was under 40?
My thoughts
a) let the probability of being under 30years be A and the probability of being under 20years be B
P(A)= 0.003+0.124+0.263+0.29
= 0.68
P(B)= 0.003+0.124
= 0.127
P(B/A)= P(B).P(A/B) / P(A)
to find P(A/B) i am now lost
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Part a)
A = event of selecting a woman less than 20 years of age
B = event of selecting a woman less than 30 years of age
Add up the values for the groups "<15 years" and "15-19 years"
P(A) = probability of event A occurring
P(A) = probability of selecting a woman less than 20 years of age
P(A) = 0.003+0.124
P(A) = 0.127
This is the probability of selecting a woman from either group
Similarly, add up the values for the groups from "<15 years" up to "25-29 years"
P(B) = 0.003+0.124+0.263+0.29
P(B) = 0.68
This is the probability of selecting a woman who is less than 30 years of age
Since A is a subset of B, this means P(A and B) = P(A). In other words, if you pick someone from group A, they are automatically in group B. Anyone under 20 years of age is also less than 30 years of age.
So, P(A and B) = 0.127
Putting this all together and we have this conditional probability:
P(A|B) = P(A and B)/P(B)
P(A|B) = 0.127/0.68
P(A|B) = 0.18676
representing the idea that if we know for sure the woman is under 30 years of age, then the probability of picking someone less than 20 years old is 0.18676 roughly
Answer: 0.18676 (which is approximate)
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Part b)
A = event of selecting a woman less than 40 years of age
B = event of selecting a woman who is 35 years or older
Add up the probabilities for the groups from "<15 years" up to "35-39 years" to capture the collection of women who are less than 40 years old.
P(A) = 0.003+0.124+0.263+0.29+0.22+0.085
P(A) = 0.985
Add up the values for the groups from "35-39 years" and upward
P(B) = 0.085+0.014+0.001
P(B) = 0.1
If we overlap the two regions for A and B, we effectively get one group: "35-39 years" which satisfies both "under 40" and "35 or older"
which is why P(A and B) = 0.085
P(A|B) = probability event A happens given B is certain to happen
P(A|B) = probability of selecting a woman who is under 40, given she is 35 or older
P(A|B) = P(A and B)/P(B)
P(A|B) = 0.085/0.1
P(A|B) = 0.85
Answer: 0.85
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