document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172452>Question 1172452</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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 .<BR>\n" );
document.write( "a. If company A gets the​ job, what is the probability that company B did not​ bid?<BR>\n" );
document.write( "b. What is the probability that company A will get the​ job? </SPAN>\n" );
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document.write( "Part (a)<BR>\n" );
document.write( "Define the events<BR>\n" );
document.write( "B = Company B bids on a job<BR>\n" );
document.write( "A = Company A gets the job contract\r<BR>\n" );
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document.write( "This leads to<BR>\n" );
document.write( "P(B) = probability that company B bids on a job<BR>\n" );
document.write( "P(A) = probability that company A gets the job contract\r<BR>\n" );
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document.write( "We know that company B bids 60% of the time, so,<BR>\n" );
document.write( "P(B) = 0.60<BR>\n" );
document.write( "P(B') = 1-P(B) = 1-0.60 = 0.40<BR>\n" );
document.write( "The B' means \"event B does not happen\".\r<BR>\n" );
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document.write( "We have these conditional probabilities<BR>\n" );
document.write( "P(A given B') = 0.50<BR>\n" );
document.write( "P(A given B) = 0.25<BR>\n" );
document.write( "that were provided in the instructions\r<BR>\n" );
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document.write( "Recall the conditional probability formula<BR>\n" );
document.write( "P(A given B) = P(A and B)/P(B)<BR>\n" );
document.write( "which rearranges to<BR>\n" );
document.write( "P(A and B) = P(A given B)*P(B)\r<BR>\n" );
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document.write( "Similarly,<BR>\n" );
document.write( "P(A and B') = P(A given B')*P(B')\r<BR>\n" );
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document.write( "So we know that<BR>\n" );
document.write( "P(A and B) = P(A given B)*P(B)<BR>\n" );
document.write( "P(A and B) = 0.25*0.60<BR>\n" );
document.write( "P(A and B) = 0.15<BR>\n" );
document.write( "we also can find,<BR>\n" );
document.write( "P(A and B') = P(A given B')*P(B')<BR>\n" );
document.write( "P(A and B') = 0.50*0.40<BR>\n" );
document.write( "P(A and B') = 0.20\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "P(A) = (A and B happens) OR (A and B' happens)<BR>\n" );
document.write( "P(A) = P(A and B) + P(A and B') ......... law of total probability<BR>\n" );
document.write( "P(A) = 0.15 + 0.20<BR>\n" );
document.write( "P(A) = 0.35\r<BR>\n" );
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document.write( "Use of a probability tree is a visual way to organize all this information.\r<BR>\n" );
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document.write( "Now we can say:<BR>\n" );
document.write( "P(B' given A) = P(B' and A)/P(A)<BR>\n" );
document.write( "P(B' given A) = P(A and B')/P(A)<BR>\n" );
document.write( "P(B' given A) = 0.20/0.35<BR>\n" );
document.write( "P(B' given A) = 20/35<BR>\n" );
document.write( "P(B' given A) = (5*4)/(5*7)<BR>\n" );
document.write( "P(B' given A) = 4/7<BR>\n" );
document.write( "This represents the probability of company B not bidding, given company A got the job.\r<BR>\n" );
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document.write( "<font color=red>Answer: 4/7</font><BR>\n" );
document.write( "Note: in decimal form, 4/7 = 0.57 approximately\r<BR>\n" );
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document.write( "=================================================<BR>\n" );
document.write( "Part (b)\r<BR>\n" );
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document.write( "This was computed earlier when we got p(A) = 0.35\r<BR>\n" );
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document.write( "It seems that it might be handy to do part (b) first, then tackle part (a). However, there may be an alternative route. \r<BR>\n" );
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document.write( "<font color=red>Answer: 0.35</font><BR>\n" );
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