/0's Methods thread

[/0]

Thought I might as well start one 

A brand of detergent is sold in bottles of two sizes - standard and large. For each size, he content (in litres) of a randomly chosen bottle is normally distributed with mean and standard deviation as given in the table:



Using the result and , find the probability that there is more detergent in four randomly chosen standard bottles than in three randomly chosen large bottles. (Assume is normally distributed)

I think this question is quite random and err.. a bit weird  thanks

[Collin Li]

We want to find:

is assumed to be a normal distribution. The parameters are:





(to 4 d.p.)


= normalcdf(0,1E99,0.01,0.04187) = 0.5944

[dekoyl]

I remember this question from the Essentials book. Even now I still don't fully understand it. Hopefully nothing like this will ever appear.

[shinny]

I've had very similar styles of questions on my SACs and such so it's worth learning how to do those sorts of questions. After one or two of them, you should have the hang of them anyhow. Worst comes to worst, stick it in your cheatbook =P

[/0]

We want to find:

is assumed to be a normal distribution. The parameters are:





(to 4 d.p.)


= normalcdf(0,1E99,0.01,0.04187) = 0.5944

Thanks coblin

[bec]

We want to find:

is assumed to be a normal distribution. The parameters are:





(to 4 d.p.)


= normalcdf(0,1E99,0.01,0.04187) = 0.5944

Sorry to hijack this thread...but why do we want to find Pr(4X - 3Y > 0)? I've never seen a question anything like this before!

[Damo17]

Sorry to hijack this thread...but why do we want to find Pr(4X - 3Y > 0)? I've never seen a question anything like this before!

find the probability that there is more detergent in four randomly chosen standard bottles than in three randomly chosen large bottles. (Assume is normally distributed)

[/0]

Yeah 4X-3Y = the difference in volume of 4 standard bottles and 3 large bottles then by finding

we are finding the probability that the difference is positive. i.e. there is more volume in the 4 standard bottles than the 3 large bottles. Since is normally distributed we can use the properties of normal distribution to find it.

[Pandemonium]

Yep, as shinjitsux said it's best to not leave any preventable possibilities of screwing yourself over open.

[bec]

Thanks, I get what you mean now. Let's hope there is a question like this in november...

[Glockmeister]

After all, you can't work out the of question appearing in exams

[/0]

2 tricky questions I had trouble with
They're pretty long though so you've been warned

1. A game of chance consists of rolling a disc of diameter 2 cm on a horizontal square board. The board is divided into 25 small squares of side 4 cm. A player wins a prize if, when a disc settles, it lies entirely within any one small square. There is a ridge round the outside edge of the board so that the disc always bounces back, cannot fall off and lies entirely within the boundary of the large square:
Prizes are awarded as follows:


When no skill is involved, the centre of the disc may be assumed to be randomly distributed over the accessible region.
a) Calculate the probability in any one throw of winning:
i) 50c
ii) 25c
iii) 12c
iv) 5c
v) no prize
b) The proprietor wishes to make a profit in the long run, but is anxious to charge as little as possible to attract customers. He charges C cents, where C is an integer. Find the lowest value of C that will yield a profit.


2. A new machine is to be developed by a manufacturing company. Prototypes are to be made until one satisfies the specifications of the company. Only then will it go into production. If, however, after three prototypes are made none are satisfactory, then the project is to be abandoned.
It is estimated that the probability of a prototype will fail to produce a satisfactory model is 0.35, independent of any other already tested.

a) It is estimated that the cost of developing and testing th efirst prototype is $7 million and each subsequent prototype developed costs half of the one before. Find the expected cost of the project.

b) If a machine is developed then it is estimated that the income will be $20 million. (If the project is abandoned there is no income.) Find the expected profit.

[bec]

What are the answers? For question 1 (and I'm pretty sure I'm wrong but here goes), I got:
a) i) 1/100
ii) 2/25
iii) 1/25
iv) 3/25
v) 3/4

b) 15 cents

2a) $8439380
b) I don't get what they mean by "income" as opposed to "profit"


(I'd show how I worked this stuff out but since it's so likely I'll be wrong I'll wait until I find out...)

[dcc]

2. A new machine is to be developed by a manufacturing company. Prototypes are to be made until one satisfies the specifications of the company. Only then will it go into production. If, however, after three prototypes are made none are satisfactory, then the project is to be abandoned.
It is estimated that the probability of a prototype will fail to produce a satisfactory model is 0.35, independent of any other already tested.

a) It is estimated that the cost of developing and testing th efirst prototype is $7 million and each subsequent prototype developed costs half of the one before. Find the expected cost of the project.

Ok, note that the question here says that 'Prototypes are to be made until one satisfies the specifications of the company'.  This means that if the first prototype is produced satisfactorily, then there is no need to make any future prototypes.

So our sample space is where denotes the event of the nth prototype succeeding, etc.

Now, using the given information & representing these events in a table (with corresponding cost):

x	()	()	()	()
Pr(X = x):	0.65	0.2275	0.079625	0.042875
Cost (*10^6):	7	10.5	12.25	13.125

So the expected cost is million.

b) If a machine is developed then it is estimated that the income will be $20 million. (If the project is abandoned there is no income.) Find the expected profit.

So now we just subtract 20 from each cost except if the project fails, and this is the loss (of course, it will be negative, so the loss is a profit)




million.

Hopefully I haven't made any LARGE GLARING ERRORS

[/0]

Thanks bec and dcc! The answers to the second question are correct. However, for the first question,

1.
a)
i)
ii)
iii)
iv)
v)

b) cents