document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982731>Question 982731</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A 4-sided die is rolled. A 1 occurs 27.9% of the time, a 2 occurs 23.5% of the time, a 3 occurs 26.6% of the time, and a 4 occurs 22.0% of the time.<BR>\n" );
document.write( "a) If the die is rolled 26 times, what is the probability that an even number occurs exactly 12 times?<BR>\n" );
document.write( "b) What is the expected value of a single roll of the die?<BR>\n" );
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document.write( "Given:<BR>\n" );
document.write( "A 4-sided die, with the following pmf:<BR>\n" );
document.write( "1: 0.279<BR>\n" );
document.write( "2: 0.235<BR>\n" );
document.write( "3: 0.266<BR>\n" );
document.write( "4: 0.220<BR>\n" );
document.write( "(Total=1.000)<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) Rolled 26 times, need probability that an even number occurs exactly 12 times.<BR>\n" );
document.write( "(b) Find expected value of a single roll of the die.<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "(A) Probability of getting an even number 12 times out of 26 tosses.<BR>\n" );
document.write( "Probability of getting an even number at each toss, <BR>\n" );
document.write( "p = P(2)+P(4)=0.235+0.220=0.455<BR>\n" );
document.write( " <BR>\n" );
document.write( "To find probability of rolling an even number 12 times out of 26, we use binomial theorem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28n%2Cr%29=C%28n%2Cr%29p%5Er%2A+%281-p%29%5E%28n-r%29 ALIGN=MIDDLE  ALT=\"P%28n%2Cr%29=C%28n%2Cr%29p%5Er%2A+%281-p%29%5E%28n-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++++where+C%28n%2Cr%29=n%21%2F%28r%21%28n-r%29%21%29 ALIGN=MIDDLE  ALT=\"++++where+C%28n%2Cr%29=n%21%2F%28r%21%28n-r%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Therefore<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2826%2C12%29=C%2826%2C12%29%2A0.455%5E12%2A+%281-0.455%29%5E%2826-12%29=0.15508 ALIGN=MIDDLE  ALT=\"P%2826%2C12%29=C%2826%2C12%29%2A0.455%5E12%2A+%281-0.455%29%5E%2826-12%29=0.15508\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( " <BR>\n" );
document.write( "(B) Expected value of a single throw E[X]<BR>\n" );
document.write( "Using the standard formula,<BR>\n" );
document.write( "E[X]=sum x*p(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A0.279%2B2%2A0.235%2B3%2A0.266%2B4%2A0.220=2.427 ALIGN=MIDDLE  ALT=\"1%2A0.279%2B2%2A0.235%2B3%2A0.266%2B4%2A0.220=2.427\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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