pprakash786 wrote:
Can someone please explain this?
you have to find the value of the following ratio:
\(ratio = \frac{P(y)}{P(x)}\)
with:
\(P(y)\): probability that the prize winner completed survey A and lived in a residence more than 75 years old at the time of the survey.
\(P(x)\): probability that the prize winner completed survey B and lived in a residence built in the previous 15 years.
We can decompose \(P(y)\) as (because of this assumption
"Assume that percent of income spent on housing is proportionally distributed across residents of housing of varying ages"):
\(P(y) = \)(probability that the prize winner completed survey A)*(probability of live in a residence more than 75 years old at the time of the survey)
also, we can do the same with \(P(x)\):
\(P(x) = \)(probability that the prize winner completed survey B)*(probability of live in a residence built in the previous 15 years)
Lets start with \(P(y)\):probability that the prize winner completed survey A (numerator: possible outcomes): \((37.5*(9\%+43\%) + 49.5*(27\%+9\%))\)
probability that the prize winner completed survey A (denominator: total outcomes): \((37.5 + 49.5)\)
probability that the prize winner completed survey A: \(\frac{(37.5*(9\%+43\%) + 49.5*(27\%+9\%))}{(37.5 + 49.5)}\)
for the second term:probability of live in a residence more than 75 years old at the time of the survey (numerator: possible outcomes): \((100\%-85\%)*132,057,804\)
probability of live in a residence more than 75 years old at the time of the survey (denominator: total outcomes): \(132,057,804\)
probability of live in a residence more than 75 years old at the time of the survey: \(\frac{(100\%-85\%)*132,057,804}{132,057,804}\)
and then:
\(P(y) = \frac{(37.5*(9\%+43\%) + 49.5*(27\%+9\%))}{(37.5 + 49.5)}*\frac{(100\%-85\%)*132,057,804}{132,057,804}\)
and for \(P(x)\):probability that the prize winner completed survey B (numerator: possible outcomes): \((37.5*(48\%) + 49.5*(64\%))\)
probability that the prize winner completed survey B (denominator: total outcomes): \((37.5 + 49.5)\)
probability that the prize winner completed survey B: \(\frac{(37.5*(48\%) + 49.5*(64\%))}{(37.5 + 49.5)}\)
for the second term:probability of live in a residence built in the previous 15 years (numerator: possible outcomes): \((15\%-0\%)*132,057,804\)
probability of live in a residence built in the previous 15 years (denominator: total outcomes): \(132,057,804\)
probability of live in a residence built in the previous 15 years: \(\frac{(15\%-0\%)*132,057,804}{132,057,804}\)
therefore, \(P(x)\):
\(P(x) = \frac{(37.5*(48\%) + 49.5*(64\%))}{(37.5 + 49.5)}*\frac{(15\%-0\%)*132,057,804}{132,057,804}\)
Lets construct the ratio (a good strategy to overcome this exercise is to do not do any calculation, because we can cancel a lot of terms!):
\(ratio = \frac{P(y)}{P(x)} = \frac{37.5*(9\%+43\%) + 49.5*(27\%+9\%)}{(37.5*(48\%) + 49.5*(64\%))} = \frac{19.5 + 17.82}{18+31.68} = \frac{37.32}{49.68} = 0.75\)
(please notice that I already have simplified the expression)Option B (2/3 = 0.66)
¿Why do I get that number?, because I picked smart numbers in order to cancel some terms (0% and 15% = 15%, and also 100%-85% = 15%), but, you can pick more accurate numbers, and you will see that the result converges to 0.66.