document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177013>Question 1177013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You roll two six-sided fair dice.\r<BR>\n" );
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document.write( "a.  Let A be the event that the first die is odd and the second is a 4, 5, or 6.<BR>\n" );
document.write( "    P(A) =  <BR>\n" );
document.write( " Round your answer to four decimal places.\r<BR>\n" );
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document.write( "b.  Let B be the event that the sum of the two dice is 10.<BR>\n" );
document.write( "    P(B) = <BR>\n" );
document.write( " Round your answer to four decimal places.\r<BR>\n" );
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document.write( "c.  Are A and B mutually exclusive events?\r<BR>\n" );
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document.write( "Yes, they are Mutually Exclusive<BR>\n" );
document.write( "No, they are not Mutually Exclusive\r<BR>\n" );
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document.write( "d.  Are A and B independent events?\r<BR>\n" );
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document.write( "They are not Independent events<BR>\n" );
document.write( "They are Independent events </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "The set of odd numbers on a single die are {1,3,5}<BR>\n" );
document.write( "This is out of six values total {1,2,3,4,5,6}<BR>\n" );
document.write( "The probability of rolling an odd number is 3/6 = 1/2.\r<BR>\n" );
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document.write( "The probability of rolling 4,5, or 6 is also 1/2 for identical steps as shown above.\r<BR>\n" );
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document.write( "The probability of rolling an odd number on the first die and one of {4,5,6} on the second die is (1/2)*(1/2) = 1/4.\r<BR>\n" );
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document.write( "Answer: P(A) = <font color=red>1/4</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Search online for \"sum of dice chart\" to see all the possible ways we can add up two values on two dice. There are 36 total possible outcomes (because 6*6 = 36). Of that total, exactly 3 of them have a sum of 10\r<BR>\n" );
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document.write( "Those 3 are:<BR>\n" );
document.write( "4+6 = 10<BR>\n" );
document.write( "5+5 = 10<BR>\n" );
document.write( "6+4 = 10\r<BR>\n" );
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document.write( "We then can say<BR>\n" );
document.write( "P(B) = (# of ways to roll a 10)/(# of ways to roll two dice)<BR>\n" );
document.write( "P(B) = 3/36<BR>\n" );
document.write( "P(B) = 1/12\r<BR>\n" );
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document.write( "Answer: P(B) = <font color=red>1/12</font><BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Mutually exclusive events are two or more events that cannot happen at the same time. For example, flipping \"heads\" and \"tails\" on the same coin and on the same toss is not possible, so we consider that mutually exclusive.\r<BR>\n" );
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document.write( "However, events A and B as defined earlier are not mutually exclusive. It is possible to have them both happen at the same time and that's when we roll (5,5); ie when we roll the two '5's.\r<BR>\n" );
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document.write( "Answer: <font color=red>No, they are <u>not</u> mutually exclusive</font>\r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "Let's assume that we are told that event B has occurred. Recall that we have 3 ways to add to ten.<BR>\n" );
document.write( "4+6 = 10<BR>\n" );
document.write( "5+5 = 10<BR>\n" );
document.write( "6+4 = 10<BR>\n" );
document.write( "This consists of the entire universe of possibilities, aka the sample space.\r<BR>\n" );
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document.write( "Of those three ways in the sample space, only one of them has the first die as odd. This is the 5+5 = 10 line, which was referenced back in part (c).\r<BR>\n" );
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document.write( "Because of this, we have a probability of 1/3 to have event A happen given that we know B has already happened. \r<BR>\n" );
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document.write( "So P(A given B) = 1/3 which is shortcut way of writing \"P(A) = 1/3 given B has occurred\".\r<BR>\n" );
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document.write( "But this contradicts P(A) = 1/4 found back in part (a). \r<BR>\n" );
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document.write( "The fact P(A) changes from 1/4 to 1/3 tells us that we do not have independent events. Events A and B are dependent events.\r<BR>\n" );
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document.write( "If events A,B were independent, then neither event would affect the others probability. This is shown in the two equations below<BR>\n" );
document.write( "P(A given B) = P(A)<BR>\n" );
document.write( "P(B given A) = P(B)<BR>\n" );
document.write( "Again, these two equations are only valid if and only if A,B are independent events.\r<BR>\n" );
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document.write( "Answer: <font color=red>They are <u>not</u> independent events.</font><BR>\n" );
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