Question 1113309
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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: <font color=red>0.18676</font> (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: <font color=red>0.85</font>

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