document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190764>Question 1190764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that we roll a pair of fair dice, so each of the 36 possible outcomes is equally likely. Let <BR>\n" );
document.write( "A denote the event that the first die lands on 3, let B be the event that the sum of the dice is 8, and <BR>\n" );
document.write( "let C be the event that the sum of the dice is 7.<BR>\n" );
document.write( "a. Are A and B independent?<BR>\n" );
document.write( "b. Are A and C independent? </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "A = the first die lands on 3<BR>\n" );
document.write( "B = the sum of the dice is 8\r<BR>\n" );
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document.write( "P(A) = 6/36 = 1/6 because there are 6 ways to roll the second die if the first die is fixed at \"3\", and this is out of 36 ways to roll two dice.\r<BR>\n" );
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document.write( "P(B) = 5/36 because there are five ways to sum to 8 as shown below<BR>\n" );
document.write( "2+6 = 8<BR>\n" );
document.write( "3+5 = 8<BR>\n" );
document.write( "4+4 = 8<BR>\n" );
document.write( "5+3 = 8<BR>\n" );
document.write( "6+2 = 8\r<BR>\n" );
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document.write( "P(A and B) = 1/36<BR>\n" );
document.write( "We have only one way to sum to 8 such that the first die is 3\r<BR>\n" );
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document.write( "Now notice that <BR>\n" );
document.write( "P(A)*P(B) = (1/6)*(5/36) = 5/216<BR>\n" );
document.write( "which does NOT match with P(A and B) = 1/36\r<BR>\n" );
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document.write( "So P(A and B) = P(A)*P(B) is a false statement\r<BR>\n" );
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document.write( "<font color=red>Therefore, the events A and B are <u>not</u> independent.</font> We consider them dependent.\r<BR>\n" );
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document.write( "Here's another way to see why A and B aren't independent.\r<BR>\n" );
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document.write( "We found earlier that<BR>\n" );
document.write( "P(A) = 1/6\r<BR>\n" );
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document.write( "If somehow we know 100% that event B happened, then we have<BR>\n" );
document.write( "P(A given B) = 1/5<BR>\n" );
document.write( "this is because the sample space (originally 36 items) has reduced to 5 items only\r<BR>\n" );
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document.write( "If we know event B happened, then the sample space is<BR>\n" );
document.write( "2+6 = 8<BR>\n" );
document.write( "3+5 = 8<BR>\n" );
document.write( "4+4 = 8<BR>\n" );
document.write( "5+3 = 8<BR>\n" );
document.write( "6+2 = 8<BR>\n" );
document.write( "i.e. the five ways to add to 8. Of those 5 ways, only one has 3 as the first item. \r<BR>\n" );
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document.write( "P(A) = 1/6 doesn't match with P(A given B) = 1/5<BR>\n" );
document.write( "We need P(A given B) = P(A) to be true if we want A and B to be independent.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "A = the first die lands on 3<BR>\n" );
document.write( "C = the sum of the dice is 7\r<BR>\n" );
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document.write( "P(A) = 1/6 was calculated earlier.\r<BR>\n" );
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document.write( "Ways to sum to 7<BR>\n" );
document.write( "1+6 = 7<BR>\n" );
document.write( "2+5 = 7<BR>\n" );
document.write( "3+4 = 7<BR>\n" );
document.write( "4+3 = 7<BR>\n" );
document.write( "5+2 = 7<BR>\n" );
document.write( "6+1 = 7<BR>\n" );
document.write( "There are 6 sums here out of 36 ways to roll two dice.<BR>\n" );
document.write( "This means P(C) = 6/36 = 1/6\r<BR>\n" );
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document.write( "event A and C = first die lands on 3 AND the sum is 7<BR>\n" );
document.write( "The only way for this to happen is to have the sum 3+4 = 7.<BR>\n" );
document.write( "Which means,<BR>\n" );
document.write( "P(A and C) = 1/36\r<BR>\n" );
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document.write( "Notice how<BR>\n" );
document.write( "P(A)*P(C) = (1/6)*(1/6) = 1/36<BR>\n" );
document.write( "which matches with the P(A and C)\r<BR>\n" );
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document.write( "We found that P(A and C) = P(A)*P(C) is a true equation.\r<BR>\n" );
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document.write( "<font color=red>Therefore, events A and C are independent.</font><BR>\n" );
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