document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1028732>Question 1028732</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Potholes requiring repair in a section of a national highway occur at an average rate of 3.2 potholes per kilometre.<BR>\n" );
document.write( "1. What is the probability that there are no potholes that required repair in 5km of the highway?<BR>\n" );
document.write( "2. What is the probability that at most three potholes require repair in 200m? </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "Potholes requiring repair in a section of a national highway occur at an average rate of 3.2 potholes per kilometre.<BR>\n" );
document.write( "1. What is the probability that there are no potholes that required repair in 5km of the highway?<BR>\n" );
document.write( "2. What is the probability that at most three potholes require repair in 200m?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "In probability problems, recognizing an appropriate distribution is half the problem solved.  The remaining part is just plugging in the parameters followed by a few key strokes on the calculator.<BR>\n" );
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document.write( "We will examine the problem with respect to the criteria for the Poisson distribution, as follows, extracted from<BR>\n" );
document.write( "http://stattrek.com/probability-distributions/poisson.aspx\r<BR>\n" );
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document.write( "A Poisson experiment is a statistical experiment that has the following properties:\r<BR>\n" );
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document.write( "The experiment results in outcomes that can be classified as successes or failures (i.e. potholes or no potholes).<BR>\n" );
document.write( "The average number of successes (&#956;) that occurs in a specified region is known. (yes, &mu;=3.2 potholes per km)<BR>\n" );
document.write( "The probability that a success will occur is proportional to the size of the region (by the context of the problem, the longer the road, more potholes will be found)<BR>\n" );
document.write( "The probability that a success will occur in an extremely small region is virtually zero.  (chances of finding a pothole in a given metre length of road is very small).<BR>\n" );
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document.write( "Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc. <BR>\n" );
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document.write( "Thus the problem satisfies all the requirements of the Poisson distribution.<BR>\n" );
document.write( "We can then calculate the required values accordingly.<BR>\n" );
document.write( "The probability is given by:<BR>\n" );
document.write( "P(x; &#956;) = (e^-&#956;) (&#956;^x) / x! <BR>\n" );
document.write( "x=given number of occurrences<BR>\n" );
document.write( "&mu;=average rate (=3.2 potholes/km)<BR>\n" );
document.write( "e=natural log base = 2.7182818284...<BR>\n" );
document.write( "n! = factorial function = n*(n-1)*....3*2*1<BR>\n" );
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document.write( "1. no potholes in 5 km.<BR>\n" );
document.write( "&mu; for 5 km = 5*3.2=16<BR>\n" );
document.write( "P(0;16)=e^(-16)*16^0/0!=e^(-16)*1/1=1.125*10^(-7)<BR>\n" );
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document.write( "2. up to 3 potholes in 200m<BR>\n" );
document.write( "&mu;=3.2*(200/1000)=0.64 potholes / 200 m<BR>\n" );
document.write( "P(x&le;3; 0.64)<BR>\n" );
document.write( "=P(x=0;0.64)+P(x=1;0.64)+P(x=2;0.64)+P(x=3;0.64)<BR>\n" );
document.write( "=0.52729+0.33747+0.10799+0.02304<BR>\n" );
document.write( "=0.99579\r<BR>\n" );
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document.write( " </SPAN>\n" );
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