Question 1172452: Construction company A is determining whether it should submit a bid for a new shopping center. In the past, their main competitor, construction company B, has submitted bids 60% of the time. If company B does not bid on a job, the probability that company A will get the job is 0.50 If company B bids on a job, the probability that company A will get the job is 0.25 .
a. If company A gets the job, what is the probability that company B did not bid?
b. What is the probability that company A will get the job?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
Define the events
B = Company B bids on a job
A = Company A gets the job contract
This leads to
P(B) = probability that company B bids on a job
P(A) = probability that company A gets the job contract
We know that company B bids 60% of the time, so,
P(B) = 0.60
P(B') = 1-P(B) = 1-0.60 = 0.40
The B' means "event B does not happen".
We have these conditional probabilities
P(A given B') = 0.50
P(A given B) = 0.25
that were provided in the instructions
Recall the conditional probability formula
P(A given B) = P(A and B)/P(B)
which rearranges to
P(A and B) = P(A given B)*P(B)
Similarly,
P(A and B') = P(A given B')*P(B')
So we know that
P(A and B) = P(A given B)*P(B)
P(A and B) = 0.25*0.60
P(A and B) = 0.15
we also can find,
P(A and B') = P(A given B')*P(B')
P(A and B') = 0.50*0.40
P(A and B') = 0.20
Therefore,
P(A) = (A and B happens) OR (A and B' happens)
P(A) = P(A and B) + P(A and B') ......... law of total probability
P(A) = 0.15 + 0.20
P(A) = 0.35
Use of a probability tree is a visual way to organize all this information.
------------------------
Now we can say:
P(B' given A) = P(B' and A)/P(A)
P(B' given A) = P(A and B')/P(A)
P(B' given A) = 0.20/0.35
P(B' given A) = 20/35
P(B' given A) = (5*4)/(5*7)
P(B' given A) = 4/7
This represents the probability of company B not bidding, given company A got the job.
Answer: 4/7
Note: in decimal form, 4/7 = 0.57 approximately
=================================================
Part (b)
This was computed earlier when we got p(A) = 0.35
It seems that it might be handy to do part (b) first, then tackle part (a). However, there may be an alternative route.
Answer: 0.35
|
|
|
