Daily question

[farha]

a,b,c are necessarily false nd d is possibly true..

[ankita]

ab and c are necesarilly false and s is possibly true.

[varun]

ans a b c are necessarily false and d is possibly true.
 

ans country a should export x and b should export y.

[Amit Goyal]

8-12 March, 2009

A cab was involved in a hit and run accident at night. Two cab companies, the green and the blue, operate in the city. You are given the following data:
a) 85% of the cabs in the city are green and 15% are blue.
b) A witness identified the cab as blue. The court tested the reliability of witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.
What is the probability that the cab involved in the accident was blue rather than green?

At each generation a microbe either splits into two perfect copies of itself or dies. If the probability of splitting is p, what is the probability that a single microbe will produce an everlasting colony?

Consider the following situation where the consumer's utility is given by u=min{x,y}.
Suppose px=py=1/2, M=1.
1) What will be the optimal consumption bundle of the utility maximizing individual?
2) Now suppose for promoting good x, company announces buy one get one free. Will that change the optimal consumption bundle of the individual in question? Will his utility increase at the new consumption bundle?
3) Now suppose individual just cares about integer amounts of good x and good y consumed in optimum and does not care about fractions. In other words his utility is given by u=min{[x],[y]} + (I/2) where I is the income left after spending on x and y.
Compare the two situations:
a) Company has an offer of buy one get one free.
b) Company offers 50% off on the purchase of good x i.e. effective price of good x is 1/4.
Suppose Marginal cost of producing good x is 1/8. Which of the above two situations is better for consumer and which one is better for the producer? Compare the profits and utility of the consumer and producer in above two cases.
[-] is the greatest integer function i.e [x] is the greatest integer less than or equal to x.
Eg--> [1.4]=1, [0.3]=0, [2]=2, [-1.3]=-2

Consider the following situation in which there is a need for designing a rent sharing mechanism for an apartment:
Assume its a two bedroom apartment and there are two individuals(1, 2). One room is bigger than the other. Lets call the bigger room B and smaller room S. The total rent of the house is Rs.10,000. Assume both individuals have the capacity to pay off the entire rent by themselves i.e. each of them have atleast Rs.10,000 in their pocket. Suppose both Individuals have the same utility function
U(B=1,S=0,R)= 7000-R, where R is the rent paid by the individual and B=1 denotes that he occupied bigger room and not the smaller room.
U(B=0,S=1,R)= 5000-R
U(B=1,S=1,R)= 9000-R
Answer the following:
1) Suggest a rent sharing scheme that ensures equal utility for both the individuals as well as both rooms are occupied. Is the allocation you suggested pareto efficient?
2) Does there exist a pareto efficient allocation in which one individual get both the rooms? If yes, give the allocation. If no, explain why?

There are 6000 commuters who commute to work from Delhi to Gurgaon. There are two ways to make the trip: one is to drive on NH8 and the other on MG road. MG road is uncongested and it takes 45 minutes to commute to work. Travel time on NH8 however, depends on the number of people: if there are N commuters, then it takes (20 + (N/100)) minutes.
(a) If commuters are free to choose their routes, how many will take MG road? How many will take NH8? Calculate the total number of person-minutes per day spent by commuters.
(b) The Delhi Government wants to minimize the total person-minutes spent on commuting. How many commuters should be allocated to NH8 and MG road respectively?

[Amit Goyal]

13-15 March, 2009

Consider the following situation of herd behavior: A person on the street deciding which of two restaurants to dine in. Suppose that both the restaurants look appealing, but both are empty because it is early evening; so at random, this person chooses restaurant A. Soon a couple walks down the same street in search of a place to eat. They see that restaurant A has customers while B is empty, and choose A on the assumption that having customers makes it the better choice. And so on as potential diners observe greater number of people eating at restaurant A they would chose for the same. Compare the situations in the following simple scenario:
Two customers (1 and 2), two restaurants(A and B).
Situation 1: Customer 1 arrives first and choose to dine in one of the restaurants with probability 1/2 each. Customer 2 arrives second and do what the first customer did. He choose A if customer 1 choose A and B if he choose B.
Situation 2: Two customers arrive simultaneously and choose to dine in one of the restaurants independently of each other with probability half each.
Problem 1: Compare the expected profits of the two restaurants in the above two situations when they make a profit of $1 per individual served.
Problem 2: Compare the expected profits of the two restaurants in the above two situations when they make a profit of $1 for the first individual and additional $2 of profit when they also manage to attract the second individual. So in this case if restaurant A serves both 1 and 2 then it makes a profit of $3. Comment on the importance of first customer for the restaurant in the above two situations.
People interested in knowing more about this kind of research might want to read the following classic paper on herd behavior by Prof. Abhijit Banerjee.
Banerjee, A, "A Simple Model of Herd Behavior," Quarterly Journal of Economics, 107 (3), pp. 797-817.

Consider the situation of a cricket match between India and Australia. India likes setting the target and Australia likes chasing the target. Compare the following options of toss to decide who will bat first:
Option 1: The toss is made as it is done presently. Referee tosses a coin and one of the captains (say, Indian captain) make a call heads or tails. If he wins he takes a decision to bat first or field first. And if he lose then the other captain gets the same privilege.
Option 2: Coin is tossed. If its head then India will bat first and if its a tail then India will field first.
Which option is better/ worse/ same for both the teams?

Suppose we say that an allocation x is Pareto optimal if and only if there does not exist an allocation, say y, such that: every individual in the economy considers y to be atleast as good as x, and atleast one individual considers y to be strictly better than x. Now, consider a society of 4 individuals. This society has five allocations to choose from a,b,c,d and e. Individual preferences over these allocations are as follows:
1st ind: a P b P c P d P e
2nd ind: b P c P d P e P a
3rd ind: c P d P e P a P b
4th ind: d P e P a P b P c

a P b P c P d P e means that the individual prefers allocation a over b, b over c, c over d...e. Find out the pareto optimal allocations for this society. Explain your answer.

[Amit Goyal]

Solution to Farha's question:

P(A wins)=a/(a+b)
P(B wins)=b/(a+b)

[ankita]

probabilty that the car is blue is 4/5.

[ankita]

1/p

[varun]

probability that car is blue is 12/ 29.

[Kanika]

prob car is blue is 12/29 =0.15*0.80/0.15*0.80+0.85*0.20

[kanika]

(1)the optimal consumption bundle is (1,1)
(2)his utility will remain constant at 1 as it is a minimum function.how evr his optimum consumtion bundle will now be (2,1).

[kanika]

(3)The producer will prefer buy one get one free of x,however the consumer will prefer 50% discount on good x.

[kanika]

(a)The number of comuters on NH-8 will be 2500 and the number  of commuters on Mg Road will be 6000-2500=3500
(b)IT takes 27000 person minutes
(b)If the Delhi Govt  wants to mimimizee thetotal number of person minutes,there should be 1250 commuters on NH-8 AND 4750 ON MG ROAD.

[Amit Goyal]

16 March 2009
Consider Ms. Bijlee whose utility function is min {E, W}, where E is her electricity consumption and W is her consumption of widgets. Suppose Ms. Bijlee’s income is 10 and the prices of widgets and electricity are 1. In order to curb Ms. Bijlee’s electricity consumption, the electricity company decides to impose a surcharge of Re. 1 on every unit of electricity consumed in excess of 4 units. What is the resulting reduction in Ms. Bijlee’s  electricity consumption?
Ref: DSE Exam 2004

[Khushboo]

1- P(car is blue) = 12/29

2- Pareto efficient allocation are only a,b,c,d but not e bcz d always does better than e. But 1st individual prefers a to any other allocation, any movement from here make him worse off.  Similarly 2nd individual prefers b to any other allocation, any movement from here make him worse off, as on.... Therefore, there is no way to make some one better off without making any one worse off.

[Amit Goyal]

17 Mar, 2009
1. Write the negation of the following:
i) I am rich and happy.
ii) John or Jim is mistaken.
iii) Mathematics and Physics are sciences.
iv) Neither of you are right.
v) If it rains today, I’ll stay home.
vi) One and only one of you may go.
vii) 7 and 14 are primes.
viii) John, Jack and Jim are 17.
ix) New York is a big city if and only if Chicago is not a big city.
x) If he I wins, I can’t lose.
xi) Both countries are at fault.

[Amit Goyal]

18-21 Mar, 2009
A survey of a group’s viewing habits over the last year revealed the following
information:
(i) 28% watched gymnastics
(ii) 29% watched baseball
(iii) 19% watched soccer
(iv) 14% watched gymnastics and baseball
(v) 12% watched baseball and soccer
(vi) 10% watched gymnastics and soccer
(vii) 8% watched all three sports.
Calculate the percentage of the group that watched none of the three sports
during the last year.

The probability that a visit to a primary care physician’s (PCP) office results in neither
lab work nor referral to a specialist is 35% . Of those coming to a PCP’s office, 30%
are referred to specialists and 40% require lab work.
Determine the probability that a visit to a PCP’s office results in both lab work and
referral to a specialist.

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an
unknown number of blue balls. A single ball is drawn from each urn. The probability
that both balls are the same color is 0.44 .
Calculate the number of blue balls in the second urn.

An auto insurance company has 10,000 policyholders. Each policyholder is classified as
(i) young or old;
(ii) male or female; and
(iii) married or single.
Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The
policyholders can also be classified as 1320 young males, 3010 married males, and 1400
young married persons. Finally, 600 of the policyholders are young married males.
How many of the company’s policyholders are young, female, and single?

[ankita]

the bundle is (1,1)

the bundle is a minimum bundle, the utility will remain same and optimal bundle will be (2,1)

[ankita]

48% watched atleast one match. so 52% watched none.

[ankita]

refered to both is .05%