Question

The Bank of Canada takes several measures to protect Canadian Economy. One of the tools Bank of Canada uses to adjust monetary policies is changing the interest rate. Increasing in interest rate would reduce how much Canadian borrow and spend. This will affect the housing market, as the increased interest rate means lower ability to borrow. A real estate advisor, however, thinks the Ottawa real estate market is hot and that an increase in interest rate would not lower the prices of houses. He assesses the likelihood of Bank of Canada’s action and its effect on house prices in Ottawa:

The likelihood of an increase in the average price of houses in Ottawa if the interest rate increases = 0.15

The likelihood of an increase in the average price of houses in Ottawa if the interest rate maintains the same level = 0.75

The chance of Bank of Canada’s maintaining the current interest rate = 0.65

We are certain that the Bank of Canada does not lower the interest rate in the near future. Bank of Canada believes to strengthen the Economy, the current interest rate should be maintained or increase. If you notice an increase in the price of houses, what probability do you assign to the event that Bank of Canada increased the interest rate?

Answer 1

Solution

Let

A1 represent the event that Bank of Canada increases the interest rate

A2 represent the event that Bank of Canada maintains the interest rate

A3 represent the event that Bank of Canada lowers the interest rate and

B represent the event that there is an increase in the average price of houses in Ottawa

Back-up Theory

If A and B are two events such that probability of B is influenced by occurrence of A, then

Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)…...…....….(1)

P(B) = {P(B/A) x P(A)} + {P(B/A^C) x P(A^C)}……………………………………………….(2)

If A is made up of k mutually and collectively exhaustive sub-events, A₁, A₂,..A_k,

P(B) = sum over i = 1 to k of {P(B/A_i) x P(A_i)} ………………………………….......…....(3)

P(A/B) = P(B/A) x {P(A)/P(B)}……………………………..…………………...........…….(4)

Now, to work out the solution,

With the terminology given at the top, the given data translate in probability language as follows:

The chance of Bank of Canada’s maintaining the current interest rate = 0.65 => P(A2) = 0.65 ……..............…… (5)

We are certain that the Bank of Canada does not lower the interest rate in the near future. => P(A3) = 0.........… (6)

Since total probability is 1, (5) and (6) => P(A1) = 0.35 ………………………………...................………………….. (7)

The likelihood of an increase in the average price of houses in Ottawa if the interest rate increases = 0.15

=> P(B/A1) = 0.15 ………………………………………………………………….......................…………………….... (8)

The likelihood of an increase in the average price of houses in Ottawa if the interest rate maintains the same level = 0.75

=> P(B/A2) = 0.75 ………………………………………....................................................................................................….... (9)

We want,

the probability of the event that Bank of Canada increased the interest rate if there is an increase in the price of houses.

The above statement, in probability language would be:

P(A1/B), which vide (4) is

= P(B/A1) x {P(A1)/P(B)}

= 90.15 x 0.35)/P(B) [vide (8) and (7)] .....………………………………………………………………………………………….. (10)

Now, vide (3),

P(B) = {P(B/A1) x P(A1)} + {P(B/A2) x P(A2)} + {P(B/A3) x P(A3)}

= (0.15 x 0.35) + (0.75 x 0.65) + (0 x 0) + [vide (8), (7), (9), (5), and (6)]

= 0.0525 +0.4875

= 0.54……………………………………………………………………………………………………………………………………. (11)

(10) and (11) =>

If there is an increase in the price of houses, probability of the event that Bank of Canada increased the interest rate

= 0.0525/0.54

= 0.0972 Answer

DONE