document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1024672>Question 1024672</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A fair coin is tossed until x heads are found in a row.  On average how many total flips will it take to get x heads in a row? </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "A fair coin is tossed until x heads are found in a row.  On average how many total flips will it take to get x heads in a row?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( " <BR>\n" );
document.write( "Take the particular case of 10 heads in a row.<BR>\n" );
document.write( "For a fair coin, on average, half the flips result in heads.<BR>\n" );
document.write( "So over 18 throws, we have 9 heads (on average).  To achieve 10 heads, the 19th throws must be a head to fulfill 10 heads, with probability of 0.5.<BR>\n" );
document.write( "Generalizing, and on average, we need 2x-1 flips to get x heads.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Looking at it in a more theoretical way, the problem can be modelled using the negative binomial distribution.<BR>\n" );
document.write( "The probability density function is given by<BR>\n" );
document.write( "P(a,n)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28a%2Bb-1%2Ca-1%29%2Ap%5Ea%2A%281-p%29%5Eb ALIGN=MIDDLE  ALT=\"C%28a%2Bb-1%2Ca-1%29%2Ap%5Ea%2A%281-p%29%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where C(a+b-1,a-1)=(a+b-1)!/((a-1)!b!)<BR>\n" );
document.write( "and <BR>\n" );
document.write( "a=number of successes<BR>\n" );
document.write( "b=number of failures <BR>\n" );
document.write( "meaning n=a+b=number of flips.<BR>\n" );
document.write( "The distribution function is the sum of the probabilities for which k=0 to b, where k is the number of failures.<BR>\n" );
document.write( "The average value of flips is such that the distribution function equals 0.5.<BR>\n" );
document.write( "For a fair coin with p=0.5, b=a-1, hence n=a+b=2a-1.<BR>\n" );
document.write( "For more detailed discussion, Google the subject, or start by reading:<BR>\n" );
document.write( "http://mathworld.wolfram.com/NegativeBinomialDistribution.html<BR>\n" );
document.write( "https://onlinecourses.science.psu.edu/stat414/node/80<BR>\n" );
document.write( "https://www.me.utexas.edu/~jensen/ORMM/computation/unit/rvadd/discrete_dist/neg_binomial.html<BR>\n" );
document.write( " </SPAN>\n" );
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