document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174461>Question 1174461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a certain city, sports bikes are being targeted by thieves. Assume that the probability of a sports bike being stolen is 0.09 while the probability is only 0.5 for a regular bike. Taking, as an approximation for all bikes in that area, the nationwide proportion 0.19 of sports bikes, find the probability that a stolen bike is a sports bike. And find the probability that a bike will be stolen. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850657 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850657\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $S$ be the event that a bike is a sports bike, and $R$ be the event that a bike is a regular bike.<BR>\n" );
document.write( "Let $T$ be the event that a bike is stolen.<BR>\n" );
document.write( "We are given:<BR>\n" );
document.write( "$P(T|S) = 0.09$ (probability a sports bike is stolen)<BR>\n" );
document.write( "$P(T|R) = 0.005$ (probability a regular bike is stolen)<BR>\n" );
document.write( "$P(S) = 0.19$ (proportion of sports bikes)<BR>\n" );
document.write( "$P(R) = 1 - P(S) = 1 - 0.19 = 0.81$ (proportion of regular bikes)\r<BR>\n" );
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document.write( "We want to find $P(S|T)$, the probability that a stolen bike is a sports bike, and $P(T)$, the probability that a bike is stolen.\r<BR>\n" );
document.write( "\n" );
document.write( "Using Bayes' theorem:<BR>\n" );
document.write( "$P(S|T) = \frac{P(T|S)P(S)}{P(T)}$\r<BR>\n" );
document.write( "\n" );
document.write( "First, we find $P(T)$, the probability a bike is stolen:<BR>\n" );
document.write( "$P(T) = P(T|S)P(S) + P(T|R)P(R)$<BR>\n" );
document.write( "$P(T) = (0.09)(0.19) + (0.005)(0.81)$<BR>\n" );
document.write( "$P(T) = 0.0171 + 0.00405$<BR>\n" );
document.write( "$P(T) = 0.02115$\r<BR>\n" );
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document.write( "Now we can find $P(S|T)$:<BR>\n" );
document.write( "$P(S|T) = \frac{P(T|S)P(S)}{P(T)}$<BR>\n" );
document.write( "$P(S|T) = \frac{(0.09)(0.19)}{0.02115}$<BR>\n" );
document.write( "$P(S|T) = \frac{0.0171}{0.02115}$<BR>\n" );
document.write( "$P(S|T) \approx 0.8085$\r<BR>\n" );
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document.write( "Therefore:<BR>\n" );
document.write( "The probability that a stolen bike is a sports bike is approximately 0.8085.<BR>\n" );
document.write( "The probability that a bike will be stolen is 0.02115.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{0.8085, 0.02115}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );