document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029807>Question 1029807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An auditor for American Health Insurance reports that 20% of policyholders submit a claim during the year. 15 policyholders are selected randomly. What is the probability that at least 3 of them submitted a claim the previous year?     </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "An auditor for American Health Insurance reports that 20% of policyholders submit a claim during the year. 15 policyholders are selected randomly. What is the probability that at least 3 of them submitted a claim the previous year?    <BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "The probability of success (submitted a claim) is p=0.20, and remains constant throughout.<BR>\n" );
document.write( "Size of sample, n=15, and policyholders are selected randomly (assumed independently as well).<BR>\n" );
document.write( "x=3, number of successes for which probability is required.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The above data satisfies the necessary conditions for modelling with the binomial distribution, which estimates the probability of x successes out of n each with a probability of p as:<BR>\n" );
document.write( "P(X=x,p,n)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cx%29%2Ap%5E%28x%29%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cx%29%2Ap%5E%28x%29%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <BR>\n" );
document.write( "x=3<BR>\n" );
document.write( "n=15<BR>\n" );
document.write( "p=0.2 and<BR>\n" );
document.write( "C(n,x)=n!/(x!(n-x)!) is the number of combinations for x objects chosen from n.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Hence<BR>\n" );
document.write( "P(X=x,p,n)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cx%29%2Ap%5E%28x%29%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cx%29%2Ap%5E%28x%29%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2815%2C3%29%2A0.2%5E3%2A0.8%5E12 ALIGN=MIDDLE  ALT=\"C%2815%2C3%29%2A0.2%5E3%2A0.8%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=455*0.008*0.0687195<BR>\n" );
document.write( "=0.25014<BR>\n" );
document.write( " <BR>\n" );
document.write( "For more explanations on the conditions required to model with binomial distributions, and more examples, see:<BR>\n" );
document.write( "http://www.euclid.host-ed.me/probability/binomialDistribution.html<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );