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September 27th, 2012, 12:46 AM   #1
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Need help to find minimal cost to complete a task

There is a task that it is either success or failure with fixed probability. We need to pay more in order to get better success rate. If there are sufficient resources, we could make sure it can be success.

Success rate (P) - Cost (C)
Code:
20% - $10
40% - $20 
60% - $40 
80% - $80 
100% - $160
As shown above, we pay $10 for 20% success rate or $160 if we need 100% success. If the task is not success, we could always retry by paying same cost until the process is success. For example,
Code:
1st trial: $80, failure
2nd trial: $20, failure
3rd trial: $$80, success
In the above example, we complete the task after 3 trial and it cost $180 in total.
We would like to find the most cost-effective solution (minimum cost) in general. Say, "Given $100, we have 90% confident to complete the task".
Is it able to do some calculation for that by providing success rate and cost? Thank you for help.
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September 27th, 2012, 01:44 AM   #2
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Re: Need help to find minimal cost to complete a task

Hey summoner.

What kind of conditions do you need? You need to specify conditions like for example what the probability needs to be (and for how many trials).

Once you do this, you have a function for the Cost given the number of trials and the possibilities and then you want to consider how to make sure that you get a minimum probability for so many successes.

In general you can have many different kinds of probabilities for each trial: for example you can assign the first with one probability, the second with another and so on.

For general probabilities you can use what is called a Probability Generating Function to get the PDF of the number of successes of arbitrary success and failure rates.
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September 27th, 2012, 02:36 AM   #3
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Re: Need help to find minimal cost to complete a task

Thanks for your reply.

The number trial is unlimited and we need to find the best solution with minimum cost.
Say assuming we have 100 task, I want a cheap solution with 80% confidence to succeed.

I will check Probability-generating function that you have suggested. Thanks.
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November 6th, 2012, 05:08 PM   #4
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Re: Need help to find minimal cost to complete a task

I would like to simplify the case by restricting the choice that a user can only choose either 1 scheme to take the task. Say for example, we can now choose the following 2 scheme. User can do a trial with same scheme until they get success

Success rate (P) - Cost (C)
Code:
2% - $1
20% - $10
There is a discussion whether a user should choose a scheme with more trials but less success rate for achieving minimal cost
For example,
Code:
1st trial: 20% success by paying $10, failure
2nd trial: 20% success by paying $10, failure
3rd trial: 20% success by paying $10, success

Cost = 3*10 = $30
Code:
1st trial: 2% success by paying $1, failure
2nd trial: 2% success by paying $1, failure
3rd trial: 2% success by paying $1, failure
4th trial: 2% success by paying $1, failure
...
20th trial: 2% success by paying $1, success

Cost = 20*1 = $20
Someone claimed that 2% scheme should be better as we have more trial and chance for success. This statement differs from mine.
It is supposed to be related to geometric series for getting 80% success in n trials.
Code:
Percent(S) = p + (1-p)*p + (1-p)^2*p + ...
Cost = C * S
I would like to get more advance. Thank you
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