document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=430376>Question 430376</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given the probabilities below:<BR>\n" );
document.write( "P(a) = 0.25<BR>\n" );
document.write( "P(B) = 0.30<BR>\n" );
document.write( "P(C) = 0.55<BR>\n" );
document.write( "P(A and C) = 0.05<BR>\n" );
document.write( "P(b and C) = 0<BR>\n" );
document.write( "P(B|A) = 0.48\r<BR>\n" );
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document.write( "A.) Are A and B independent events? Why or Why not?<BR>\n" );
document.write( "B.) Find P(A and B)<BR>\n" );
document.write( "C.) Find P (A or B)<BR>\n" );
document.write( "D.) Find P(A|B)<BR>\n" );
document.write( "E.) Are B and C mutually exclusive events?  Why or why not?<BR>\n" );
document.write( "F.) Are B and C independent events? Provide mathematical proof. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #782303 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=782303\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The tutor @robertb has a great solution. I don't have much, if anything, to add really except for the fact that if A and B were independent, then,\r<BR>\n" );
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document.write( "P(A | B) = P(A)<BR>\n" );
document.write( "P(B | A) = P(B)\r<BR>\n" );
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document.write( "Both must be true for A and B to be independent. We're told that P(B) = 0.30 and P(B | A) = 0.48 which contradicts the second equation shown above. \r<BR>\n" );
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document.write( "Put another way: If A and B were independent, then either event would not alter the probability of the other. The notation P(B | A) indicates that we are given event A has happened. Saying P(B|A) = 0.48 is the same as saying \"the probability event B happens is 0.48 given A has happened\". But this is altered from P(B) = 0.30; therefore event A has changed the probability of event B's occurrence. Ultimately the two events are not independent.\r<BR>\n" );
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document.write( "All of this explanation applies to part A only<BR>\n" );
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