# Probability and Subcontractors

• TheClincher
In summary, the contractor is able to get at least one of the two subcontractors 90% of the time. The probability that both subcontractors will be available to do the next job is .80.
TheClincher

## Homework Statement

A contractor has two subcontractors for his excavation work. Experience shows that in 60% of the time, subcontractor A was available to do a job, whereas subcontractor B was available 80% of the time. Also, the contractor is able to get at least one of these two subcontractors 90% of the time.
a) Probability that both subcontractors will be available to do the next job.
b) If contractor learned that subcontractor A is not available for the job, what is the probability that the other subcontractor will be available?

## Homework Equations

A = sub A available
B = sub B available

## The Attempt at a Solution

a) P(AB) = P(A)P(B)

b is the one I have a question on. I think it's the following set up but I don't know how to continue or what to do with it:
b) P(B|$$\bar{A}$$) = P(B$$\bar{A}$$)/P($$\bar{A}$$)

Okay uhh I don't know if this is sensible but I took P(B$$\bar{A}$$) and did the following

P(B$$\bar{A}$$) = 1-complement of P(B$$\bar{A}$$) and expanded it to get 0.32. So if I plug that into P(B|$$\bar{A}$$) = P(B$$\bar{A}$$)/P($$\bar{A}$$) I got 0.32/0.40 = 0.80. Eh? Can someone steer me right

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I think you are making a mistake in assuming that the availability of the two contractors is independent. This makes P(AB)=P(A)P(B) wrong. Draw a Venn diagram type thing for the availability space. P(A) is 0.6, P(B) is 0.8 and the region outside of P(A) or P(B) is 0.1. Reason from there. What's the correct value for P(AB)?

Oh, so, hmm... P(AB) = P(A) + P(B) - P(A U B) = 0.50?

TheClincher said:
Oh, so, hmm... P(AB) = P(A) + P(B) - P(A U B) = 0.50?

That's what I get. Now how about the second part?

(I'll denote complements like this: c{A} for complement of A)

B is continuing to vex me. From the problem statement, I'm interpreting this: P(c{A}B U Ac{B}) = P(c{A}B) + P(Ac{B}) = 0.90; that is, if one is not available the contractor still has a 90% chance of being able to hire the other subcontractor. I think I have to expand the two elements P(c{A}B) + P(Ac{B}), but I don't think I know how. :-\

Try drawing a picture first. It really helps. Draw a circle A which represents .6, a circle B that represents .8, outside of them both represents .1. From a) we know the overlap between A and B is .5. That leaves .1 in A outside of B and .3 in B outside of A. Now b) is easy. The total area outside of A is .4, B covers .3 of that. So the odds are? Now try and translate that picture into your set notation.

ah well I've got the solution and the concept. drawing venn diagrams certainly does help. i spent last night focusing on a purely graphical approach instead of a symbolic-computation approach on a different probability problem and solved it successfully. thank you for your guidance, Dick.

So I have drawn the venn diagram and I understand what you are saying about the areas, but I still don't quite understand what area constitutes my answer. I want to assume that the two, A and B, are statistically independent, which would give .8, but i know that's too easy. A little more help please...

The area you want for a) is the area of the intersection of A and B. Let O be the area of the intersection of A and B. Now solve for all of the areas. DON'T assume they are statistically independent. Even if they were, you still wouldn't get .8.

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P(AUB) = .9, P(AB) = .5; the compliment of A is .4, if the compliment of A occurs, that leaves .6 insides the sample space, right? With .1 of that being the intersection of compliment of A and compliment of B, so that leaves .5 left, but that doesn't make sense looking at the venn diagram.
Am I still taking the wrong approach?

helpme74 said:
P(AUB) = .9, P(AB) = .5; the compliment of A is .4, if the compliment of A occurs, that leaves .6 insides the sample space, right? With .1 of that being the intersection of compliment of A and compliment of B, so that leaves .5 left, but that doesn't make sense looking at the venn diagram.
Am I still taking the wrong approach?

Sorry, I should have said intersection instead of union in my last post. I corrected it. And, yes, P(AB)=.5. My venn diagram makes sense P(A)=.6=.5+.1, P(B)=.8=.3+.5. What part of yours doesn't make sense?

I mean when looking at the diagram: if the compliment of A occurs, that means that A will not occur, and P(AB) will also not occur; that just leaves .3 left inside of B, and I'm getting .5 for the answer to the question, not .3

helpme74 said:
I mean when looking at the diagram: if the compliment of A occurs, that means that A will not occur, and P(AB) will also not occur; that just leaves .3 left inside of B, and I'm getting .5 for the answer to the question, not .3

Which question? Are you trying answer b) or a)?

Please forgive me for not clarifying earlier, I'm trying to answer (b). I know that .5 is the answer for (a), but I'm getting .5 and .3 for answers for (b).

I got .5 by going this route:
P(AUB) = .9, P(AB) = .5; the compliment of A is .4, if the compliment of A occurs, that leaves .6 insides the sample space, right? With .1 of that being the intersection of compliment of A and compliment of B, so that leaves .5 left for my answer.

And I got .3 by looking at the diagram and trying to solve visually:
When looking at the diagram: if the compliment of A occurs, that means that A will not occur, and P(AB) will also not occur; that just leaves .3 left inside of B for my answer.

This seems rather simple but I am still confused.

helpme74 said:
Please forgive me for not clarifying earlier, I'm trying to answer (b). I know that .5 is the answer for (a), but I'm getting .5 and .3 for answers for (b).

I got .5 by going this route:
P(AUB) = .9, P(AB) = .5; the compliment of A is .4, if the compliment of A occurs, that leaves .6 insides the sample space, right? With .1 of that being the intersection of compliment of A and compliment of B, so that leaves .5 left for my answer.

And I got .3 by looking at the diagram and trying to solve visually:
When looking at the diagram: if the compliment of A occurs, that means that A will not occur, and P(AB) will also not occur; that just leaves .3 left inside of B for my answer.

This seems rather simple but I am still confused.

If you are trying to do b) then you exclude the region where A is available. That leaves you with an area of .3 in B and and area of .1 outside of B, right? What's the PROBABILITY (not area) that B is available?

Ah, yes. I think perhaps I 'over-complicated' things.
Thank you VERY much for your assistance, sir. I am indebted.

## What is probability?

Probability is a measure of the likelihood of an event occurring. It is typically expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

## How is probability used in relation to subcontractors?

In the context of subcontractors, probability is used to assess the chances of a particular subcontractor meeting project requirements, completing work on time, and staying within budget. It can also be used to evaluate the likelihood of encountering issues or delays with subcontractors.

## What factors influence the probability of success with subcontractors?

The probability of success with subcontractors can be influenced by a variety of factors, including the subcontractor's experience and track record, their level of communication and collaboration with the project team, the complexity of the work being subcontracted, and the overall project management and planning processes in place.

## How can probability be calculated for subcontractors?

There are various methods for calculating the probability of success with subcontractors, including using historical data and past performance, conducting risk assessments, and gathering input from project team members and experts in the field. It is important to regularly reassess and update these calculations as the project progresses.

## What are some strategies for managing probability with subcontractors?

To effectively manage probability with subcontractors, it is important to establish clear expectations and communication channels, regularly monitor and track subcontractor progress, address any issues or concerns promptly, and have contingency plans in place for potential delays or issues. It is also important to maintain open and honest communication with subcontractors and foster a collaborative working relationship.

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