# Thread: could you please solve this problem

1. ## could you please solve this problem

I propose to contract a taxi ﬁrm for a daily journey of 3km. Study of the traﬃc conditions has suggested to me that the variable time of the journey is reasonably described by the following model:
No. of minutes 5 6 7 8 9 10
Probability 0.08 0.13 0.33 0.25 0.14 0.07 .

(a) The contract oﬀered to me by one ﬁrm involves a charge of £1 per km and 30 pence per minute. What is the average cost of a journey? What is the standard deviation of the cost?
(b) If I envisage the contract running for one year (regarded as 300 independent working days), what would I expect to pay in total? What is the standard deviation of the total annual cost? Do you consider the assumption of independence between days reasonable? Explain your answer.
(c) A second ﬁrm oﬀers a contract of £1.5 per km and 10 pence per minute. Which contract will be more advantageous to me in the long run?

2. ## Re: could you please solve this problem

Have you made no attempt at all to do this yourself? First find the cost of each possible trip. For example, with the first company the cost is £1 per km and the trip is 3 km so there is a base charge of £3. There is a charge of 30p per minute so if the trip take 5 min, the charge will be £3 plus 5(30)p= £4.50. If the trip takes 6 minutes the charge will be £3 plus 6(30) p= £4.80 etc. To find the average cost, multiply each trip charge by the probability of that trip: there is a probability 0.08 the trip will take 5 minutes: 0.08(4.50)= £0.36, there is a probability 0.13 the trip will take 6 minutes: 0.13(4.80)= £0.624, etc. Add all of those to get the average cost. To find the standard deviation, subtract that average cost from each of the costs, then square. Again multiply each by the probability the trip takes that many minutes and add. Finally take the square root to find the standard deviation.

With 300 trips per year, multiply the average cost per trip by 300 to get the total cost for a year. The standard deviation per year is the standard deviation times the square root of 300.

Do the same for the other company: the cost is £1.5 per km and 10p per minute so a 3 km trip taking 5 minutes would cost 3(1.5)+ 5(.10)= 4.5+ 0.50= £5.00. A 3km trip taking 6 minutes would cost 3(1.5)+ 6(.10)= 4.50+ 0.60= £5.50, etc. Determining which is the better deal only requires looking at the average cost per day, not the total cost per year or the standard deviation.