Probability help question complication

[sara.]

Hey! So my teachers not very good at explaining things and in the end it usually leaves me more confused...kinda leaving me like this http://www.youtube.com/watch?v=6-lRDMOprfA
Anyways I just came across AN now when I was searching for some help on google haha I've kinda made a list of questions I need help on anything will be appreciated
 
Probability revision complication (need help)

In order to assess the quality of computer games being produced, the manufacturer selects 10 games at random before they are put into boxes for delivery to shops, and inspects them.
1   If it is known that 8% of games are defective, find, correct to four decimal places:
a   the probability that one of the games in the sample will be defective
b   the probability that at least one of the games in the sample will be defective
c   the probability that exactly one of the games in the sample will be defective, given that at least one of the games is defective
d   the expected number of defective games in the sample
e   the standard deviation of the number of defective games in the sample

2   It is found after a new machine is installed that the probability that exactly one of the games in a sample of size 10 being defective, given that at least one of the games is defective is equal to 0.56. What proportion of defective games is now being produced? (Give your answer correct to two decimal places.)

Suppose that there are two doctors in a country town. Records show that from year to year:
•   of Dr Arnold’s patients, there is a probability of a that a patient stays with him, and thus of (1 – a) that the patient will move to Dr Brain
•   of Dr Brain’s patients, there is a probability of b that a patient stays with him, and thus of (1 – b) that the patient will move to Dr Arnold.

1   Write down the transition matrix T which can be used to represent this information.

2   At the beginning of the year 55% of the patients go to Dr Arnold and 45% go to Dr Brain. Find (in terms of a and b):
a   the percentage of patients estimated to be at each doctor at the end of year 1
b   the percentage of patients estimated to be at each doctor in the long term

3   Suppose that 10 000 patients are treated by the two doctors in the town each year. Neither doctor can afford to operate if the number of patients seen falls below 4000 annually.
a   Suppose that a = 0.8. Find a range of value of b such that each doctor can afford to operate in the long term. Give your answer correct to three decimal places.
b   Suppose that initially there are 5000 patients who go to Dr Arnold and 5000 who go to Dr Brain. If a = 0.75 and b = 0.85, will either of the doctors go out of business, and if so, who and after how many years?
 

The time it takes to complete a task (in hours) is a random variable X with probability density function
(see attached photo I don't know how to add formulas yet :s)
where c is a constant.
1   Find the value of c.

2   Sketch the graph of this distribution.

3   Write down the most likely time taken to complete the task.

4   Find the probability that the time taken to complete the task will be:
a   more than 0.8 hours
b   between 0.4 and 0.8 hours

[sara.]

anyone?? 

[SocialRhubarb]

1
a.) So we want one game to be defective, and 9 games to be normal. However, any one of the ten games could be defective, and so we multiply it by ten.

The probability of this is .

b.) The probably that at least one game is defective is the same as the probability that not all the games are working.



c.) The probability that one game exactly one gave is defective given that at least one of the games is defective is given by the probability of one divided by the probability of the other, or

d.) Expected number of 'successes' in a binomial distribution is np, where n is the number of trials and p is the probability of success. A binomial distribution is just a probability function with only two results. In this case, a game is either defective or working so we can tell it is binomial. The number of trials is 10 and the probability of success is 0.08.



e.) The variance of a binomial distribution is given by np(1-p), which in this case is equal to 0.736.

The standard deviation is the square root of the variance, and so is equal to 0.8579. The variance and standard deviation basically just tell you how spread out a particular set of data is. The higher the variance and standard deviation, the more spread out a set of data is.

2.) We do exactly what we did in part c, but instead of using the probability which we were given, we'll call the probability 'p'.



I don't have my calculator with me, but if you solve that, you should get a value for p, where p is the probability that a game is defective.

[Stevensmay]

The last part, with given equations.
1. Area under the graph of a probability density function must always equal 1. Therefore

Solving for c on a calculator yields approximately .6944
3. 'Most likely' is the time with the highest probability. We can find this by graphing f(x) and finding its maximum.
This maximum occurs at x=.6 with y = 2.5.
4. a.
Evaluating
b.
4. Evaluating with the c value found earlier.

Some part of the matrix one. I am even more unsure about this one, could someone please verify what I have done?
https://www.dropbox.com/s/2pqjoa72u2cmpee/IMAG0127.jpg

[sara.]

Spoiler

1
a.) So we want one game to be defective, and 9 games to be normal. However, any one of the ten games could be defective, and so we multiply it by ten.

The probability of this is .

b.) The probably that at least one game is defective is the same as the probability that not all the games are working.



c.) The probability that one game exactly one gave is defective given that at least one of the games is defective is given by the probability of one divided by the probability of the other, or

d.) Expected number of 'successes' in a binomial distribution is np, where n is the number of trials and p is the probability of success. A binomial distribution is just a probability function with only two results. In this case, a game is either defective or working so we can tell it is binomial. The number of trials is 10 and the probability of success is 0.08.



e.) The variance of a binomial distribution is given by np(1-p), which in this case is equal to 0.736.

The standard deviation is the square root of the variance, and so is equal to 0.8579. The variance and standard deviation basically just tell you how spread out a particular set of data is. The higher the variance and standard deviation, the more spread out a set of data is.

2.) We do exactly what we did in part c, but instead of using the probability which we were given, we'll call the probability 'p'.



I don't have my calculator with me, but if you solve that, you should get a value for p, where p is the probability that a game is defective.

THANKYOU!! Your explanations are so good, and clear! 

Spoiler

The last part, with given equations.
1. Area under the graph of a probability density function must always equal 1. Therefore

Solving for c on a calculator yields approximately .6944
3. 'Most likely' is the time with the highest probability. We can find this by graphing f(x) and finding its maximum.
This maximum occurs at x=.6 with y = 2.5.
4. a.
Evaluating
b.
4. Evaluating with the c value found earlier.

Thank you so much!!! Your answer was great also

Some part of the matrix one. I am even more unsure about this one, could someone please verify what I have done?
https://www.dropbox.com/s/2pqjoa72u2cmpee/IMAG0127.jpg

I did what you did as well, It's on the right track but not quite I've attached the answer to this post. I just need help getting there haha

[Stevensmay]

Getting the transition matrix right would have made things alot easier.

So for 2.a we want to find out what percentage of patients each doctor has. We can find this using the equation
where T is our transition matrix and is our initial conditions, in this case .55 and .45.

Thus we get Multiply these by 100 to get a percent.

In our initial conditions we had Dr A listed in row 1 and Dr B listed in row 2. Thus our result gives the percent for Dr A in row 1 and the percent for Dr B in row 2.

For part b we simply want to find what values we will get in the long run. This can be done by simply finding but can be time consuming. Another way to do this is with simple algebra. I am not exactly sure how to explain this sorry.


To find the long term average of this generic matrix is simply . This will find us the average for the thing associated with position b. In our question it will be patients at Dr Arnold's clinic.
For Dr Brain it will be the 'reverse,' effectively swapping the two cells. Thus we get people.

Question 3.
For this question we are interested in making sure that the numbers of patients seen by each doctor is at least 4000 in the long term. 'Long term' indicates we can use what we found in 2.b here, which we shall.
For Arnold, the number of patients he sees is represented by   percent of patients he gets.
Thus,

For Brain it is

We simply put these into our calculator as a simultaneous solve and find that which is our range of acceptable b values for both clinics to remain operational.

Question 3 b.
I am not sure how to do this one algebraically, but it is relatively quick to do by trial and error (bad method).
Essentially just increment n until the number of patients at a clinic drops below 4000. This happens relatively quickly, so makes me think the question was in fact meant to be approached this way. If anyone has a better solution would be great.
 

By the time we get to we have found that Dr Arnold's clinic only has 3912 patients in that year, and so will have closed.

Hopefully what I have written makes sense.

[sara.]

Getting the transition matrix right would have made things alot easier.

So for 2.a we want to find out what percentage of patients each doctor has. We can find this using the equation
where T is our transition matrix and is our initial conditions, in this case .55 and .45.

Thus we get Multiply these by 100 to get a percent.

In our initial conditions we had Dr A listed in row 1 and Dr B listed in row 2. Thus our result gives the percent for Dr A in row 1 and the percent for Dr B in row 2.

For part b we simply want to find what values we will get in the long run. This can be done by simply finding but can be time consuming. Another way to do this is with simple algebra. I am not exactly sure how to explain this sorry.


To find the long term average of this generic matrix is simply . This will find us the average for the thing associated with position b. In our question it will be patients at Dr Arnold's clinic.
For Dr Brain it will be the 'reverse,' effectively swapping the two cells. Thus we get people.

Question 3.
For this question we are interested in making sure that the numbers of patients seen by each doctor is at least 4000 in the long term. 'Long term' indicates we can use what we found in 2.b here, which we shall.
For Arnold, the number of patients he sees is represented by   percent of patients he gets.
Thus,

For Brain it is

We simply put these into our calculator as a simultaneous solve and find that which is our range of acceptable b values for both clinics to remain operational.

Question 3 b.
I am not sure how to do this one algebraically, but it is relatively quick to do by trial and error (bad method).
Essentially just increment n until the number of patients at a clinic drops below 4000. This happens relatively quickly, so makes me think the question was in fact meant to be approached this way. If anyone has a better solution would be great.
 

By the time we get to we have found that Dr Arnold's clinic only has 3912 patients in that year, and so will have closed.

Hopefully what I have written makes sense.

Thanks heaps that makes a lot of sense  

[sara.]

Sorry guys 1 more question, these mixed review one's always stump me because I never know what rules to use anyway:
 
The gestation time (the time elapsed between conception and birth) for pregnancies without problems in humans is approximately normally distributed with a mean of 266 days and a standard deviation of 16 days.
1   What percentage of pregnancies last between 234 and 298 days?
2   In a paternity case, a woman claims her pregnancy lasted 312 days. The supposed father of the child had been sent away on a scientific expedition to Antarctica 312 days before the birth. What percentage of gestation times exceed 312 days in length?
3   Suppose that 100 chosen women give birth one day at a hospital. What is the probability, to four decimal places, that at least one of the women has a gestation period of more than 312 days in length?
4   Find, to the nearest day, the length of gestation that is exceeded by 5% of pregnancies.

Thanks heaps it's great that you guys help people like me

[Stevensmay]

1. The values given are two standard deviations either side of the mean, which is 95% (from standard deviation rule). Means that 95% of all pregnancies fall between these dates.

2. Did this using the normalCDF function on my calculator. Lower bound of 312, upper bound of infinity and the mean/s.d given in the question. On the TI-nspire this would be normCdf(312, infinity symbol, 266, 16). Answer of .00202

3. Again on CAS, this time with the binomial CDF function. binomCdf(100, 0.00202, 1, 100) giving an answer of 0.1831. Can also be done by finding the probability that none of them will have a gestation period longer than 312 days, and subtracting this from 1.

4. More CAS. Inverse Normal function used. invNorm(.95, 266, 16) giving an answer of 292 days.
We want to find the x value that includes 95% of the area under the graph, giving us the .95.

Not too good at explaining the CAS ones sorry.

[sara.]

Thanks heaps again dude!! For this question :

4. More CAS. Inverse Normal function used. invNorm(.05, 266, 16) giving an answer of 240 days.
We want to find the x value that includes 5% of the area under the graph, giving us the .05.
Not too good at explaining the CAS ones sorry.

I get what you did but apparently the answer is 292 days 

[Stevensmay]

Fixed, that's what happens when you think 5=95.

[academicbulimia]

Is there any way to do these questions by hand?

[Stevensmay]

I do not know of any manual methods for these sorry.

[academicbulimia]

I do not know of any manual methods for these sorry.

It's alright I was just wondering haha