document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969421>Question 969421</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A fair sided coin is tossed 5 times\r<BR>\n" );
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document.write( "1) How many outcomes in the sample space<BR>\n" );
document.write( "2) what is the probability that the third toss is heads given that that the first toss is heads<BR>\n" );
document.write( "3) Let A be the event that the first toss is heads and B be the event that the third toss is heads. Are A and B independent, why or why not? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592324 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are 2 outcomes in the first toss, 4 in the second  HH  HT TH  TT; 8 in the third, 16 in the 4th, and 32 in the 5th.  Pascal's Triangle<BR>\n" );
document.write( "1 1<BR>\n" );
document.write( "1 2 1<BR>\n" );
document.write( "1 3 3 1<BR>\n" );
document.write( "1 4 6 4 1<BR>\n" );
document.write( "1 5 10 10 5 1<BR>\n" );
document.write( " It doesn't show the triangular form formatted here, but notice how the top two add to the bottom one.\r<BR>\n" );
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document.write( "There are 32 outcomes, if order matters, which it appears to in this question. \r<BR>\n" );
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document.write( "Given that the first toss is heads, the probability the third toss is heads is 50%.  They are independent.  Whatever happened on the first toss has no effect on the third toss.\r<BR>\n" );
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document.write( "Tree diagram hard to make here<BR>\n" );
document.write( "HT<BR>\n" );
document.write( "HH HT TH TT<BR>\n" );
document.write( "HHH HHT HTH HTT THH THT TTH TTT<BR>\n" );
document.write( "The first toss is heads (4), and you can see that the third toss is heads 50% of the time.\r<BR>\n" );
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document.write( "If independent P(outcome 1)* P(outcome 2)= P (outcome 1 and outcome 2)<BR>\n" );
document.write( "P (1st toss is heads)=50%   Probability third toss is heads is 50%.  The first AND third should be heads 25% of the time.  You can see 2 of the 8 above are, so they are independent. </SPAN>\n" );
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