The plan for a new housing development call for the construction of a large number of residential homes.A marketing expert tells the owner that because of the poor location of development,20 of the houses which are sold will be sold at a loss.A financial consultant asserts 10% of the houses will be sold at a loss.the building contractor states that only 5% of the houses will be sold at a loss.
I.if the owner feels that the market consultant is 5 times as reliable as the building contractor,what prior probabilities should the owner assign to these percentages.
II.if 8 houses are actually built and sold,and 2 of the houses are sold at a loss,what posterior probabilities should the owner assign to the 3 percentages?
In order to assign prior probabilities to the three percentages, we can use the principle of reliability. Let's denote the reliability of the marketing expert's prediction as R1 and the reliability of the building contractor's prediction as R2. Since the owner believes that the market consultant is 5 times as reliable as the building contractor, we can say that R1 = 5*R2.
Now, let's assign prior probabilities to these percentages.
Let P1 be the prior probability assigned to the marketing expert's prediction (20 houses sold at a loss),
P2 be the prior probability assigned to the financial consultant's prediction (10% houses sold at a loss),
and P3 be the prior probability assigned to the building contractor's prediction (5% houses sold at a loss).
Since probabilities must sum to 1, we have the following equation:
P1 + P2 + P3 = 1 ---(Equation 1)
We are given that the marketing expert's prediction is 5 times as reliable as the building contractor's prediction. This can be expressed as:
R1 = 5 * R2
Since reliability is the inverse of the error rate, we can write this relationship as:
1 - P1 = 5 * (1 - P3) ---(Equation 2)
We can solve the system of equations (Equation 1 and Equation 2) to find the prior probabilities assigned to the percentages.
Moving on to part II of the question, where 8 houses are actually built and sold, and 2 of the houses are sold at a loss. We need to find the posterior probabilities assigned to the three percentages given this new information.
Let's denote the posterior probabilities of the marketing expert's prediction, financial consultant's prediction, and building contractor's prediction as P1', P2', and P3' respectively.
To calculate these posterior probabilities, we can use Bayes' theorem:
P1' = (P1 * P(sold at a loss | prediction by marketing expert)) / P(sold at a loss)
P2' = (P2 * P(sold at a loss | prediction by financial consultant)) / P(sold at a loss)
P3' = (P3 * P(sold at a loss | prediction by building contractor)) / P(sold at a loss)
Here, P(sold at a loss | prediction by marketing expert), P(sold at a loss | prediction by financial consultant), and P(sold at a loss | prediction by building contractor) can be calculated based on the given information in the problem.
Please provide the specific values for P(sold at a loss | prediction by marketing expert), P(sold at a loss | prediction by financial consultant), and P(sold at a loss | prediction by building contractor), and I will be able to calculate the posterior probabilities in part II of the question.