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?
I. To determine the prior probabilities, we need to consider the reliability of the market consultant and the building contractor according to the owner's perception.
Let's assume:
P(MC) as the prior probability of the market consultant's assertion (10% of houses sold at a loss).
P(BC) as the prior probability of the building contractor's statement (5% of houses sold at a loss).
Given that the owner perceives the market consultant as 5 times as reliable as the building contractor, we can set up the following equation:
P(MC) = 5 * P(BC)
We also know that the sum of the probabilities should be 1:
P(MC) + P(BC) = 1
Substituting the value of P(MC) from the first equation into the second equation, we get:
5 * P(BC) + P(BC) = 1
Simplifying the equation, we have:
6 * P(BC) = 1
To solve for P(BC), we divide both sides by 6:
P(BC) = 1/6
Now, substituting the value of P(BC) into the first equation, we find:
P(MC) = 5 * (1/6) = 5/6
Therefore, the prior probabilities assigned to these percentages are:
P(MC) = 5/6
P(BC) = 1/6
II. To calculate the posterior probabilities, we will use Bayes' theorem. Let's assume:
P(MC|2) as the posterior probability of the market consultant's assertion given that 2 houses are sold at a loss.
P(BC|2) as the posterior probability of the building contractor's statement given that 2 houses are sold at a loss.
Using Bayes' theorem, the formula is:
P(A|B) = (P(B|A) * P(A)) / P(B)
For the market consultant:
P(MC|2) = (P(2|MC) * P(MC)) / P(2)
For the building contractor:
P(BC|2) = (P(2|BC) * P(BC)) / P(2)
P(2|MC) represents the probability of 2 houses being sold at a loss given the market consultant's assertion. Similarly, P(2|BC) represents the probability of 2 houses being sold at a loss given the building contractor's statement.
Now, let's calculate the values of P(2|MC), P(2|BC), and P(2).
Given that the market consultant asserts 10% of houses will be sold at a loss, P(2|MC) can be calculated using the binomial distribution:
P(2|MC) = (8 choose 2) * (0.10^2) * (0.90^6) = 28 * 0.01 * 0.531441 = 0.14976568
Similarly, the building contractor states that only 5% of houses will be sold at a loss, so P(2|BC) can be calculated using the binomial distribution:
P(2|BC) = (8 choose 2) * (0.05^2) * (0.95^6) = 28 * 0.0025 * 0.73509189 = 0.05774013
To calculate P(2), we need to consider all possible scenarios of selling 2 houses at a loss. This includes when the market consultant's assertion is true and the building contractor's statement is false, and vice versa.
P(2) = P(2|MC) * P(MC) + P(2|BC) * P(BC)
= 0.14976568 * (5/6) + 0.05774013 * (1/6)
≈ 0.12480592
Now, substituting these values into the formulas for the posterior probabilities:
P(MC|2) = (0.14976568 * (5/6)) / 0.12480592
P(BC|2) = (0.05774013 * (1/6)) / 0.12480592
Calculating these values will give the owner the posterior probabilities assigned to the 3 percentages.