document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089547>Question 1089547</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One day, eleven babies are born at a hospital. Assuming each baby has an equal chance of being a boy or a girl, what is the probability that at most nine of the eleven babies are girls? </SPAN>\n" );
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document.write( "One day, eleven babies are born at a hospital. Assuming each baby has an equal chance of being a boy or a girl, what is the probability that at most nine of the eleven babies are girls?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "We will first check if this problem can be modelled using the binomial distribution.<BR>\n" );
document.write( "1. Bernoulli trials, i.e. exactly two possible outcomes (girl or boy)<BR>\n" );
document.write( "2. Number of trials (n) is known before and constant throughout the experiment, i.e. independent of outcomes. (n=11)<BR>\n" );
document.write( "3. All trials are independent of each other. (assumed from context).<BR>\n" );
document.write( "4. Probability (p) of success is known, and remain constant throughout trials. (p=0.5)<BR>\n" );
document.write( "Since all criteria are satisfied, we can model this situation with binomial distribution, where the probability of x successes out of N trials each with probability of success p is given by<BR>\n" );
document.write( "P(x)=C(N,x)(p^x)(1-p)^(N-x)<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "C(N,x) is number of combinations of selecting x objects out of N.<BR>\n" );
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document.write( "n=11<BR>\n" );
document.write( "p=0.5<BR>\n" );
document.write( "x<=9<BR>\n" );
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document.write( "P(x<=9)=1-(P(x=10)+P(x=11)<BR>\n" );
document.write( "=1-(C(11,10)0.5^10*0.5^1+C(11,11)0.5^11*0.5^0)<BR>\n" );
document.write( "=1-(0.005371+0.000488)<BR>\n" );
document.write( "=0.994141\r<BR>\n" );
document.write( "\n" );
document.write( "The probability that at most 9 out of 11 newborns are girls is 0.994141, assuming boys and girls are equally probable. </SPAN>\n" );
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