Question

Q4. Police in district ABC The police force in the district ABC prepares itself to receive on a typical day, on average 12 reports of car accidents. Other cases are resolved without police being involved. On a typical day, there are 40 cases of car accidents (reported or not) What is the probability that individuals report their car accident to the police on a typical day? (2 a- points) b- What is the variability (Standard deviation) that police should expect in the number of cases reported to the police on a typical day? (2 points) c How likely is that the police does not receive any report in a day that there are 5 car accident? (assume the probability of reporting does not change) (2 points) Q5. Bank of Canada 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 rate65 We are ceain that Bank of Canada does not lower interest rate in near future, Bank of Canada believes 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? Q5. Google Google gets 3.5 billion searches in a day. In fact, it is the most used search engine around the world. In Canada, 72.06% of Canadians use Google as their main search engine. Suppose you select 10 individuals at random. a- What is the probability that at least one of them does not use Google as their main search engine? (3 points) b How many individuals, on average, you need to survey before finding the first person who prefers search engines other than Google? (3 points) c What is the probability that exactly three individuals do not use Google among 10 individuals randomly selected (3 points) d- What assumptions you have made to carry on with your calculations? (2 points)

Answer 1

4)

a) P(A car accident gets reported) = 12/40 = 0.3

b) The no. of cases getting reported follows a binomial distribution with p = 0.3 and n = 40

Hence, variance of no. of case reported = n*p*(1-p) = 40*0.3*0.7 = 8.4

=> Std deviation = sqrt(8.4) = 2.898 ~ 3

c) P(No case reported out of 5 accidents) = * 0.3⁰ * 0.7⁵ = 0.168

5) Let A = Event that Bank increased interest rate, B = Event that House prices increased

Using Baye's theorem:

P(Bank increased interest rate / House prices increased)

= P(A/B)

= P(B/A) * P(A) / ( P(B/A) * P(A) + P(B/A^c) * P(A^c) ), where A^c is the event that bank keeps interest rate the same

But, P(B/A) = 0.15, P(B/A^c) = 0.75

And, P(A) = 0.65, P(A^c) = 1 - 0.65 = 0.35

Plugging in the above values, we get

P(Bank increased interest rate / House prices increased) = 0.271

6)  

P(A Canadian uses Google as main search engine) = 0.7206

a) P(At least one of the 10 doesn't use Google as main search engine)

= 1 - P(all of them use Google as main search engine)

= 1 - 0.7206¹⁰ = 0.962

b) No. of individuals needed to survey before finding the first one who prefers search engine other than google follows a geometric distribution with p = 1 - 0.7206 = 0.2794 (probability of finding someone who prefers a search engine other than google). Let this variable be Y

Hence, E(Y) = (1-p) / p = 2.579

c) P(Exactly 3 do not use Google among the 10 individuals)

= * 0.2794³ * 0.7206⁷ = 0.264

10 3

d) The assumptions were that the event that the different individuals do not use or use Google as search engine are independent of each other. Also there are only 2 outcomes (prefer or do not prefer Google as main search engine) and no. of trials (individuals) is fixed (10 above).

6

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