document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197173>Question 1197173</A>: <I><SPAN STYLE=\"word-break: break-all;\">  When two balanced dice are rolled, 36 equally likely outcomes are possible. Let X denote the sum of the dice. a) What are the possible values of the random variable X ? b) Use random-variable notation to represent the event that the sum of the dice is 7. c) Find P(X = 7). d) Find the probability distribution of X . Leave your probabilities in fraction form. e) Construct a probability histogram for X. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830319 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=830319\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "X = sum of the two dice\r<BR>\n" );
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document.write( "Assuming we have two standard six-sided dice, each labeled 1 through 6, then the possible values of X are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 <BR>\n" );
document.write( "due to this sum of dice chart here<BR>\n" );
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document.write( "<tr><TD >+</TD><TD ><font color=red>1</font></TD><TD ><font color=red>2</font></TD><TD ><font color=red>3</font></TD><TD ><font color=red>4</font></TD><TD ><font color=red>5</font></TD><TD ><font color=red>6</font></TD></tr><tr><TD ><font color=blue>1</font></TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD></tr><tr><TD ><font color=blue>2</font></TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD></tr><tr><TD ><font color=blue>3</font></TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD></tr><tr><TD ><font color=blue>4</font></TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD></tr><tr><TD ><font color=blue>5</font></TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD><TD >11</TD></tr><tr><TD ><font color=blue>6</font></TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD><TD >11</TD><TD >12</TD></tr>\n" );
document.write( "</TABLE>Each value in black is a sum of a red and blue value.<BR>\n" );
document.write( "Eg: <font color=blue>1</font> + <font color=red>6</font> = 7 in the top right corner.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Here are all the possible ways to get a sum of 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( "I'm not exactly sure what your teacher is referring to when mentioning \"Use random-variable notation\". \r<BR>\n" );
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document.write( "It's possible that your teacher wants you to write P(X = 7) for the answer here.\r<BR>\n" );
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document.write( "I recommend getting a second opinion from your teacher, another classmate, or another tutor.\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "In the previous part, there were 6 ways to get a sum of 7<BR>\n" );
document.write( "This is out of 36 total outcomes<BR>\n" );
document.write( "6/36 = 1/6\r<BR>\n" );
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document.write( "P(X = 7) = 1/6\r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "We'll follow the same idea as part (c), but do so for X = 2 all the way up to X = 12\r<BR>\n" );
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document.write( "P(X = 2) = 1/36 since there is one way to get a sum of 2 out of 36 total outcomes\r<BR>\n" );
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document.write( "P(X = 3) = 2/36 since there are two ways to get a sum of 3 out of 36 total outcomes\r<BR>\n" );
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document.write( "P(X = 4) = 3/36 since there are three ways to get a sum of 4 out of 36 total outcomes\r<BR>\n" );
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document.write( "And so on until reaching P(X = 7) = 6/36<BR>\n" );
document.write( "So far the numerators have increased: 1,2,3,4,5,6\r<BR>\n" );
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document.write( "Then we have a countdown of numerators: 5,4,3,2,1 when going through P(X = 8) to P(X = 12)\r<BR>\n" );
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document.write( "Here's what the probability distribution looks like<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >2</TD><TD >1/36</TD></tr><tr><TD >3</TD><TD >2/36</TD></tr><tr><TD >4</TD><TD >3/36</TD></tr><tr><TD >5</TD><TD >4/36</TD></tr><tr><TD >6</TD><TD >5/36</TD></tr><tr><TD >7</TD><TD >6/36</TD></tr><tr><TD >8</TD><TD >5/36</TD></tr><tr><TD >9</TD><TD >4/36</TD></tr><tr><TD >10</TD><TD >3/36</TD></tr><tr><TD >11</TD><TD >2/36</TD></tr><tr><TD >12</TD><TD >1/36</TD></tr>\n" );
document.write( "</TABLE>I'll leave the fractions un-reduced so that each has a denominator of 36. This way things remain consistent and you can compare each P(X) value.\r<BR>\n" );
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document.write( "If your teacher requires you to reduce each fraction, then be sure to do so.<BR>\n" );
document.write( "Here is each reduction<BR>\n" );
document.write( "2/36 = 1/18<BR>\n" );
document.write( "3/36 = 1/12<BR>\n" );
document.write( "4/36 = 1/9<BR>\n" );
document.write( "6/36 = 1/6\r<BR>\n" );
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document.write( "Part (e)\r<BR>\n" );
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document.write( "Refer to the table in part (d)<BR>\n" );
document.write( "Each P(X) value tells us how tall the bar will be for any particular X value. \r<BR>\n" );
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document.write( "Example: The first bar is 1/36 units tall since P(X) = 1/36 when X = 2.\r<BR>\n" );
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document.write( "This is what the histogram would look like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/fNqWAFR.png\"><BR>\n" );
document.write( "This distribution is symmetric since the left half is a mirror copy of the right half.<BR>\n" );
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