document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158324>Question 1158324</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi Professor:\r<BR>\n" );
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document.write( "I have a question related to the probability, a stock price is currently at $10, what will be the probability the stock price will be above $10 after 5 days? Assume the stock price will be randomly moved according to the following exact percentages: -3%, -2%, -1%, 1%, 2% and 3% (so there are total 6 different possible percentages movement and the  movement are purely random among each day). \r<BR>\n" );
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document.write( "My approach to this problem is the following: there are total 6 different movement, and the total trials are 5 days, so the total permutations will be 6^5=7776, but I just can't continue the rest due to my limited math knowledge, could you help me out on this?\r<BR>\n" );
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document.write( "Thank you for reading my email and I am looking forward to hearing from you soon. \r<BR>\n" );
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document.write( "Alex <BR>\n" );
document.write( "California\r<BR>\n" );
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document.write( "Hi Tutors:\r<BR>\n" );
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document.write( "Thank you for answering my questions above (at Answer 781251 ). \r<BR>\n" );
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document.write( "I think the answer (781251) just solved one of the permutation, what if the percentage change were  repeatable negative percentages, these percentage changes are all independent, so it could have like: -3% every single days, or -3% for 3 days then another 2 days -2%, etc.  I wanted to know what are the probability under all the permutations, what are the chances that the stock price at the end of 5th day will be above $10?\r<BR>\n" );
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document.write( "again, thanks<BR>\n" );
document.write( "Alex  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #781245 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=781245\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "original response deleted... I'm looking at this further....<br><BR>\n" );
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document.write( "In my earlier attempt at solving this, I tried to simplify the problem by treating all the percent increases and decreases as being relative to the original price.  That way, for instance,  increases of 3% and 3% would exactly balance decreases of 1%, 2%, and 3%.<br><BR>\n" );
document.write( "However, that is probably not how the problem was intended.<br><BR>\n" );
document.write( "If the percentage increases or decreases are treated as multipliers, there is NO permutation of 5 of the 6 posssible percentage changes that results in an ending price exactly equal to the starting price.<br><BR>\n" );
document.write( "So I looked at the problem again....<br><BR>\n" );
document.write( "Certainly the other tutor missed the point of the problem, finding the percent increase or decrease if each of the changes is applied once.<br><BR>\n" );
document.write( "It appears to me that a purely analytic solution would be extremely tedious, making it necessary to examine each permutation of 5 of the 6 percentage changes.<br><BR>\n" );
document.write( "So I built an excel spreadsheet with all 6^5=7776 permutations of 5 of the 6 and identified the ones that produced a product greater than 1.<br><BR>\n" );
document.write( "ANSWER: 3588 of the 7776 permutations of 5 of the 6 percentage changes produce a product greater than 1.  Therefore, the probability that the stock price will be above the original $10 after 5 days is 3588/7776, or about 46.142%.<br><BR>\n" );
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