Word problem yuck!

B

blackpanther

New member
Joined
Jan 1, 2009
Messages
7
Two antibiotics are available as treatment for a common ear infection in children.
-Antibiotic A is known to effectively cure the infection 60% of the time. Treatment with antibiotic A costs $50.
-Antibiotic B is known to effectively cure the infection 90% of the time. Treatment with antibiotic B costs $80.


The antibiotics work independently of one another. Both antibiotics can be safely administered to children. A health insurance company intends to recommend one of the following two plans of treatment for children with this ear infection.
-Plan 1: Treat with antibiotic A first. If it is not effective, then treat with antibiotic B.
-Plan 2: Treat with antibiotic B first. If it is not effective, then treat with antibiotic A.

a) If a doctor treats a child with an ear infection using Plan 1, what is the probability that the child will be cured? If a doctor treats a child with an ear infection using plan 2, what is the probability that the child will be cured?

b) Compute the expected cost per child when plan I is used for treatment.
Compute the expected cost per child when plan II is used for treatment.

c) Based on the results in parts (a) and (b) which plan would you recommend? Explain. ( I guess if you can help me make it through a and b, ill do this one :p)


I have been working on this for a little while and dont really know where to start. Any help I would greatly appreciate.
 
blackpanther said:
Two antibiotics are available as treatment for a common ear infection in children.
-Antibiotic A is known to effectively cure the infection 60% of the time. Treatment with antibiotic A costs $50.
-Antibiotic B is known to effectively cure the infection 90% of the time. Treatment with antibiotic B costs $80.


The antibiotics work independently of one another. Both antibiotics can be safely administered to children. A health insurance company intends to recommend one of the following two plans of treatment for children with this ear infection.
-Plan 1: Treat with antibiotic A first. If it is not effective, then treat with antibiotic B.
-Plan 2: Treat with antibiotic B first. If it is not effective, then treat with antibiotic A.

a) If a doctor treats a child with an ear infection using Plan 1, what is the probability that the child will be cured? If a doctor treats a child with an ear infection using plan 2, what is the probability that the child will be cured?
Click to expand...

If \(\displaystyle p_a\) is the probability that A is effective, and \(\displaystyle p_b\) the probability that B is effective, then for plan 1:

\(\displaystyle p_{cure}=p_a+(1-p_a)p_b\)

and for plan 2:

\(\displaystyle p_{cure}=p_b+(1-p_b)p_a\)

That is if the first treatment is succesfull stop otherwise treat with the other.


b) Compute the expected cost per child when plan I is used for treatment.
Compute the expected cost per child when plan II is used for treatment.
Click to expand...

If \(\displaystyle c_a\) is the cost of treatment A and \(\displaystyle c_b\) the cost of treatment B the the expected cost of plan 1 is:

\(\displaystyle E(cost)=c_ap_a + (c_a+c_b)(1-p_a)\)

That is if the first treatment works we incur just the cost of that treatment, otherwise we incur the sum of the costs of both treatments (independent of if the second works or not).

You should be able to calculate the expected cost of plan 2.
 
blackpanther said:
Two antibiotics are available as treatment for a common ear infection in children.
-Antibiotic A is known to effectively cure the infection 60% of the time. Treatment with antibiotic A costs $50.
-Antibiotic B is known to effectively cure the infection 90% of the time. Treatment with antibiotic B costs $80.


The antibiotics work independently of one another. Both antibiotics can be safely administered to children. A health insurance company intends to recommend one of the following two plans of treatment for children with this ear infection.
-Plan 1: Treat with antibiotic A first. If it is not effective, then treat with antibiotic B.
For this plan to work -
A works (60% and spent $50)

A does not work (40% and spent $50) - then B will work (90%) - total spent $130

So

probability of Plan A to work = 0.6 + 0.4*0.9 = 0.96

probability of Plan A to not-work = 0.04

Expected cost = 0.6*50 + 0.36*130 + 0.04*130= 30+ 52 = 82

-Plan 2: Treat with antibiotic B first. If it is not effective, then treat with antibiotic A.

Follow the steps shown above.

a) If a doctor treats a child with an ear infection using Plan 1, what is the probability that the child will be cured? If a doctor treats a child with an ear infection using plan 2, what is the probability that the child will be cured?

b) Compute the expected cost per child when plan I is used for treatment.
Compute the expected cost per child when plan II is used for treatment.

c) Based on the results in parts (a) and (b) which plan would you recommend? Explain. ( I guess if you can help me make it through a and b, ill do this one :p)


I have been working on this for a little while and dont really know where to start. Any help I would greatly appreciate.
Click to expand...
Please show us work, indicating exactly where you are stuck - so that we know exactly where we need to begin to help you.

General - you beat me about 1 minute and I did not see yourresponse. Since I wrote all these up - I decided to put it up anyway.
 
general i like the equations u used, but im not really sure how to apply them.

We just learned this and this is a question my tecaher assigned us over the weekend. I already appreciate your help ( and urs too khan ) But could you please help me further?
 
blackpanther said:
But could you please help me further?
Click to expand...

It is now your turn to show your work.

I have shown you how to use general's equation for part (1) of your problem. Did you use paper/pencil to write down our responses - instead of just staring at those?

You should be able to use the exact same process - as shown for part (1) - for part (2) and show us your work.

Just saying "I don't understand" will not work. Exactly where you stop understanding....which term fails to penetrate?
 
B works - 90% costs 80$
B doesn't work - 10% and costs $80
A will work 60%

So

Probability (plan 2 to work) = .90(.10x.60)=.96
Probability (plan 2 not work) = .04

(hmmm? same answer i hope im not doing this wrong)

expected cost of plan 2 - .9x80+.06x130+.04x130 = $85

Correct me if im wrong, if im right, you made this really easy, and possible can you shove me into the next direction :D ?
 
blackpanther said:
B works - 90% costs 80$
B doesn't work - 10% and costs $80
A will work 60%

So

Probability (plan 2 to work) = .90 + (.10x.60)=.96
Probability (plan 2 not work) = .04

(hmmm? same answer i hope im not doing this wrong)

expected cost of plan 2 - .9x80+.06x130+.04x130 = $85

Correct me if im wrong, if im right, you made this really easy, and possible can you shove me into the next direction :D ?
Click to expand...

As far as I can tell, you have done this correctly.
 
What were the questions? - did you answer those?

What were the instructions? - did you follow those?

You should be able to (make a check list - if necessary) above questions and decide accordingly.
 
You must log in or register to reply here.
Top