document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086770>Question 1086770</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the past 111 years, a certain state suffered 32 direct hits from major(category 3 to 5) hurricanes. Assume that this was typical and the number of hits per year follows a Poisson distribution. \r<BR>\n" );
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document.write( "What is the probability that the state will not be hit by any major hurricanes in a single& year?\r<BR>\n" );
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document.write( "What is the probability that the state will be hit by at least one major hurricane in a single year?\r<BR>\n" );
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document.write( " What is the probability that the state will be hit by at least three major hurricanes in a single& year, as happened several years ago?\r<BR>\n" );
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document.write( "What is the probability that the state will be hit by at least two major hurricanes in a single year, as happened last year?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700997 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700997\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "For the past 111 years, a certain state suffered 32 direct hits from major(category 3 to 5) hurricanes. Assume that this was typical and the number of hits per year follows a Poisson distribution. <BR>\n" );
document.write( "(a) What is the probability that the state will not be hit by any major hurricanes in a single& year?<BR>\n" );
document.write( "(b) What is the probability that the state will be hit by at least one major hurricane in a single year?<BR>\n" );
document.write( "(c) What is the probability that the state will be hit by at least three major hurricanes in a single& year, as happened several years ago?<BR>\n" );
document.write( "(d) What is the probability that the state will be hit by at least two major hurricanes in a single year, as happened last year?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Given Poisson distribution, assuming historical trends up to the last century still apply today.<BR>\n" );
document.write( "Mean number of direct hits per year, &lambda; = 32/111<BR>\n" );
document.write( "PMF(k,&lambda;)=&#955;^k (e^(-&#955;)/k!<BR>\n" );
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document.write( "(a) k=0<BR>\n" );
document.write( "P(k=0)=&#955;^k (e^(-&#955;)/k!<BR>\n" );
document.write( "=32/111^0(e^(-32/111))/0!<BR>\n" );
document.write( "=1*(e^(-32/111))/1<BR>\n" );
document.write( "=e^(-32/111)<BR>\n" );
document.write( "=0.7495<BR>\n" );
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document.write( "(b) k>=1<BR>\n" );
document.write( "P(k)=1-P(k=0)=1-0.7495=0.2505<BR>\n" );
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document.write( "(c) P(k>=3)=1-(P(k=0)+P(k=1)+P(k=2))<BR>\n" );
document.write( "=1-(0.7495+0.2161+0.0311)<BR>\n" );
document.write( "=1-0.9967<BR>\n" );
document.write( "=0.0033<BR>\n" );
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document.write( "(d) P(k>=2)=1-(P(k=0)+P(k=1))<BR>\n" );
document.write( "=1-(0.7495+0.2161)<BR>\n" );
document.write( "=1-0.9656<BR>\n" );
document.write( "=0.0344 </SPAN>\n" );
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