document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=569692>Question 569692</A>: <I><SPAN STYLE=\"word-break: break-all;\"> another friends ask you to explain how to tell if two events are dependent or independent ? how would you reply? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367424 by </B><A HREF=/tutors/aboutme.mpl?userid=JBarnum><B>JBarnum(2146)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=JBarnum><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=JBarnum><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=367424\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Dependent and Independent Events<BR>\n" );
document.write( "The occurrence of some events may affect the probability of occurrence of others. For example, the complementary events A and A cannot occur simultaneously. If one took place the other is out of the game:\r<BR>\n" );
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document.write( "  P(A|A) = 0  \r<BR>\n" );
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document.write( "regardless of the probability P(A). (P(A|B) denotes the conditional probability of A assuming B.) We say that the event A is not independent of the event A (assuming P(A) &#8800; 0 of couse.) And, in general, an event A is not independent of an event B iff P(A) &#8800; P(A|B), i.e., if occurrence of B affects the probability of A. It may not be obvious right away, but the relationship \"not independent\" is symmetric: if A is not independent of B then also B is not independent of A. Formally,\r<BR>\n" );
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document.write( "  if P(A) &#8800; P(A|B) then P(B) &#8800; P(B|A).  \r<BR>\n" );
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document.write( "To see why that is so we invoke the defintion of conditional probability,\r<BR>\n" );
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document.write( "  P(A|B) = P(A&#8745;B) / P(B),  \r<BR>\n" );
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document.write( "so that P(A) = P(A|B) implies P(A&#8745;B) / P(B) = P(A), or\r<BR>\n" );
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document.write( "  P(A&#8745;B) = P(A) P(B),  \r<BR>\n" );
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document.write( "which also may serve as the definition of independency of A from B. But the later relationship is symmetric! It implies\r<BR>\n" );
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document.write( "  P(B) = P(A&#8745;B)/P(A) = P(B|A),  \r<BR>\n" );
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document.write( "which exactly means that B is independent of A. We see that two events A and B are either both dependent or independent one from the other. The symmetric definition of independency is this\r<BR>\n" );
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document.write( "(*) P(A&#8745;B) = P(A) P(B).  \r<BR>\n" );
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document.write( "Two events A and B are independent iff that condition holds. They are dependent otherwise.\r<BR>\n" );
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document.write( "It's a frequent misconception that the independency or dependency of two events relates to their having or not having an empty intersection. The case of A and its complement A supplies a clear example of two dependent events with empty intersection.\r<BR>\n" );
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document.write( "Another example was introduced in the discussion of conditional probabilities, where we considered the sets\r<BR>\n" );
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document.write( "  &#937; = {1, 2, 3, 4, 5, 6, 7, 8}, <BR>\n" );
document.write( "A = {1, 3, 5, 7}, <BR>\n" );
document.write( "B = {7, 8}, <BR>\n" );
document.write( "A+ = {1, 2, 3, 5, 7} and <BR>\n" );
document.write( "A- = {3, 5, 7}. <BR>\n" );
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document.write( "We found that P(B) = 1/4 whilst\r<BR>\n" );
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document.write( "  P(B|A+) = 1/5, <BR>\n" );
document.write( "P(B|A) = 1/4, <BR>\n" );
document.write( "P(B|A-) = 1/3. <BR>\n" );
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document.write( "What this says is that B and A are independent whereas B is dependent on both A+ and A-. (Note that B&#8745;A+ = B&#8745;A = B&#8745;A- = {7}.)\r<BR>\n" );
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document.write( "Returning to the question of survival and life expectancy, AN is the event of a new-born reaching the age of N years. By the meaning of it, for N > M, AN&#8834;AM and therefore AN&#8745;AM = AN. In particular this means that\r<BR>\n" );
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document.write( "  P(AN&#8745;AM) = P(AN)  \r<BR>\n" );
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document.write( "so that P(AN&#8745;AM) = P(AN)·P(AM) is equivalent to P(AN) = P(AN)·P(AM) which is only possible if P(AN) = 0 or P(AM) = 1. Since the latter is unrealistic and the former is unlikely for a reasonable age N, the two events AN and AM can't be independent. In fact, the probability of reaching a certain age grows with aging. As the time goes by, the size of the surviving population goes down while the number of person in a certain (advanced) age group is fixed. Thus the probability (which is the ratio of the two quantities) of getting into that group is indeed increasing with age.\r<BR>\n" );
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document.write( "in reference to: http://www.cut-the-knot.org/Probability/IndependentEvents.shtml<BR>\n" );
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