document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1020882>Question 1020882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5) Alice is trying to send a message to Bob. Eve is an eavesdropper who randomly corrupts words sent by Alice. Because of Eve, any word sent by Alice reaches Bob correctly with a probability, p, independent of other words. Alice sends messages to Bob such that each message is 2 words long. They also exchange some code language so that if any 1 of the two words sent by Alice is received correctly by Bob, then Bob can understand the message correctly. If a message is not received correctly by Bob, he indicates that to Alice and she re-sends the message. This is continued until Bob receives the message correctly.<BR>\n" );
document.write( "a)What is the probability (in terms of p) that a message sent by Alice is received incorrectly by Bob?<BR>\n" );
document.write( "b)Call your solution of Problem # 5a) as q. What should be the minimum value of q if Alice wants to give Bob a 98% guarantee that she will deliver a message correctly to Bob in 2 attempts or less?<BR>\n" );
document.write( "c) What should be the minimum value of p so that you can achieve the value of q you obtained in Problem # 5b)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #636692 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=636692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "5) Alice is trying to send a message to Bob. Eve is an eavesdropper who randomly corrupts words sent by Alice. Because of Eve, any word sent by Alice reaches Bob correctly with a probability, p, independent of other words. Alice sends messages to Bob such that each message is 2 words long. They also exchange some code language so that if any 1 of the two words sent by Alice is received correctly by Bob, then Bob can understand the message correctly. If a message is not received correctly by Bob, he indicates that to Alice and she re-sends the message. This is continued until Bob receives the message correctly.<BR>\n" );
document.write( "a)What is the probability (in terms of p) that a message sent by Alice is received incorrectly by Bob?<BR>\n" );
document.write( "b)Call your solution of Problem # 5a) as q. What should be the minimum value of q if Alice wants to give Bob a 98% guarantee that she will deliver a message correctly to Bob in 2 attempts or less?<BR>\n" );
document.write( "c) What should be the minimum value of p so that you can achieve the value of q you obtained in Problem # 5b)?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "(a)<BR>\n" );
document.write( "n=2<BR>\n" );
document.write( "probability of correct word = p<BR>\n" );
document.write( "probability of incorrect word = (1-p)<BR>\n" );
document.write( "probability of incorrect (2-word) message, q = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-p%29%5E2 ALIGN=MIDDLE  ALT=\"%281-p%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "A guarantee is a guarantee (100%).<BR>\n" );
document.write( "I suppose what is meant is what is the value of q such that 98% of the time, in the long run, a message will be received correctly in 2 attempts or less.<BR>\n" );
document.write( "We need q^2 (both attempts fail) <=2%.<BR>\n" );
document.write( "so q<=sqrt(1-98%)=0.14142 (approximately).<BR>\n" );
document.write( "so <i>maximum</i> value of q so that the message will be received correctly in 2 attempts or less is sqrt(.02)=sqrt(2/100)=sqrt(2)/10.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(c)The minimum value of p is such that<BR>\n" );
document.write( "(1-p)^2<=sqrt(2)/10<BR>\n" );
document.write( "Solve for p:<BR>\n" );
document.write( "1-p<=sqrt(sqrt(2)/10)<BR>\n" );
document.write( "p>=1-sqrt(sqrt(2)/10)<BR>\n" );
document.write( "=1-0.3761<BR>\n" );
document.write( "=0.6239 (approximately)<BR>\n" );
document.write( " <BR>\n" );
document.write( "Check:<BR>\n" );
document.write( "((1-0.6239)^2)=0.02, ok.\r<BR>\n" );
document.write( "\n" );
document.write( "Also,<BR>\n" );
document.write( "in order that you get answers as fast as possible, I suggest you post new questions instead of in the messages.  You are welcome to alert me to the question in your message if you so wish. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );