document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=997817>Question 997817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A fair coin is flipped ten times. What is the probability of the coin landing heads up exactly six times? </SPAN>\n" );
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document.write( "Given:<BR>\n" );
document.write( "A fair coin is flipped 10 times.<BR>\n" );
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document.write( "Need:<BR>\n" );
document.write( "Probability of heads coming up exactly six times.<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Typical use of the binomial distribution where:<BR>\n" );
document.write( "- probability (1/2) is constant throughout experiement<BR>\n" );
document.write( "- multi-step Bernoulli experiement (each with two possible outcomes)<BR>\n" );
document.write( "- all steps are independent of each other and random.\r<BR>\n" );
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document.write( "events and probability for each trial:<BR>\n" );
document.write( "H=heads, P(H)=1/2<BR>\n" );
document.write( "T=tails, P(T)=1/2\r<BR>\n" );
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document.write( "p, P(H) = probability of heads as outcome in each trial\r<BR>\n" );
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document.write( "Probability of exactly r events out of n is given by the binomial formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28r%2Cn%2Cp%29+=+nCr+%28p%5Er%29+%281-p%29%5E%28n-r%29 ALIGN=MIDDLE  ALT=\"P%28r%2Cn%2Cp%29+=+nCr+%28p%5Er%29+%281-p%29%5E%28n-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In the given situation, n=10, r=6, p=1/2, nCr=10C6=10!/(6!4!)=210<BR>\n" );
document.write( "so <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%0D%0AGiven%3A%0D%0AA+fair+coin+is+flipped+10+times.%0D%0A+%0D%0ANeed%3A%0D%0AProbability+of+heads+coming+up+exactly+six+times.%0D%0A+%0D%0ASolution%3A%0D%0ATypical+use+of+the+binomial+distribution+where%3A%0D%0A-+probability+%281%2F2%29+is+constant+throughout+experiement%0D%0A-+multi-step+Bernoulli+experiement+%28each+with+two+possible+outcomes%29%0D%0A-+all+steps+are+independent+of+each+other+and+random.%0D%0A%0D%0Aevents+and+probability+for+each+trial%3A%0D%0AH=heads%2C+P%28H%29=1%2F2%0D%0AT=tails%2C+P%28T%29=1%2F2%0D%0A%0D%0Ap%2C+P%28H%29+=+probability+of+heads+as+outcome+in+each+trial%0D%0A%0D%0AProbability+of+exactly+r+events+out+of+n+is+given+by+the+binomial+formula%3A%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=P%2528r%252Cn%252Cp%2529%2B=%2BnCr%2B%2528p%255Er%2529%2B%25281-p%2529%255E%2528n-r%2529+ALIGN=MIDDLE++ALT=%22P%2528r%252Cn%252Cp%2529%2B=%2BnCr%2B%2528p%255Er%2529%2B%25281-p%2529%255E%2528n-r%2529%22++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3E%0D%0A%0D%0AIn+the+given+situation%2C+n=10%2C+r=6%2C+p=1%2F2%2C+nCr=10C6=10%21%2F%286%214%21%29=210%0D%0Aso+%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >P(6,10,1/2)=(210)* ((1/2)^6) * ((1-1/2)^4) =210/1024 = 105/512\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "Probability of getting exactly 6 heads out of ten tosses of a fair coin is 105/512. </SPAN>\n" );
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