document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1018774>Question 1018774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that a particular physician determines that based on his experience with certain cancers, only 20% survive as long as 4 months following diagnosis. What is the probability that 2 out of 10 patients survive the initial 4 months following diagnosis?  Show your work.\r<BR>\n" );
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document.write( " B (k;n;p) =<BR>\n" );
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document.write( "Question:<BR>\n" );
document.write( "Suppose that a particular physician determines that based on his experience with certain cancers, only 20% survive as long as 4 months following diagnosis. What is the probability that 2 out of 10 patients survive the initial 4 months following diagnosis?  Show your work.\r<BR>\n" );
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document.write( " B (k;n;p) =<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Binomial distribution is applicable if all of the following conditions are met:<BR>\n" );
document.write( "1. Each trial is Bernoulli (success or failure), random, and independent of each other.<BR>\n" );
document.write( "2. The number of trials is fixed and known.<BR>\n" );
document.write( "3. The probability of success is known and remains constant throughout the trials.<BR>\n" );
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document.write( "For the given problem, all the conditions are met, so the binomial distribution is applicable.<BR>\n" );
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document.write( "The probability of k successes out of n trials each with a probability of success p is given by<BR>\n" );
document.write( "B(k;n;p)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Ck%29%2Ap%5Ek%2A%281-p%29%5E%28n-p%29 ALIGN=MIDDLE  ALT=\"C%28n%2Ck%29%2Ap%5Ek%2A%281-p%29%5E%28n-p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where C(n,k)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21%2F%28k%21%2A%28n-k%29%21%29 ALIGN=MIDDLE  ALT=\"n%21%2F%28k%21%2A%28n-k%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "with<BR>\n" );
document.write( "p=0.2; k=2; n=10;<BR>\n" );
document.write( "P(2;10;0.2)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2810%2C2%29%2A0.2%5E2%2A%280.8%29%5E8 ALIGN=MIDDLE  ALT=\"C%2810%2C2%29%2A0.2%5E2%2A%280.8%29%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.302 ALIGN=MIDDLE  ALT=\"0.302\"   DISABLED_style= \"max-width:90%; height:auto;\" >(approximately)<BR>\n" );
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document.write( "Answer: The probability of exactly 2 out of 10 patients survive the initial four months is 0.302.<BR>\n" );
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