document.write( "Question 892720: Use Tchebycheff's inequality to find how many times a fair coin must be tossed in order that the probability that the ratio of the number of heads to the number of tosses will lie between 0.45 and 0.55 will be atleast 0.95 \n" ); document.write( "

Algebra.Com's Answer #540755 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
Tchebycheff's inequality states that $\"P%28abs%28X+-+mu%29%3C=k%2Asigma%29%3E=1-1%2Fk%5E2\"$ , where $\"mu\"$ is the mean of the random variable and $\"sigma\"$ is its standard deviation.
\n" ); document.write( "Incidentally, if X = ratio of number of heads to n tosses of a fair coin, then the mean $\"mu+=+1%2F2\"$ , and variance is $\"sigma%5E2+=+1%2F%284n%29\"$ . (Remember, E(X) = p, while $\"Var%28X%29+=+%28pq%29%2Fn\"$ .)\r
\n" ); document.write( "\n" ); document.write( "==> standard deviation is $\"sigma+=+1%2F%282%2Asqrt%28n%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Now from the given $\"P%280.45%3C=X%3C=0.55%29%3E=0.95\"$ \r
\n" ); document.write( "\n" ); document.write( "<==> $\"P%28abs%28X+-+0.50%29%3C=0.05%29%3E=0.95\"$ \r
\n" ); document.write( "\n" ); document.write( "Now let $\"0.05+=+k%2Asigma\"$ , and $\"1+-+1%2Fk%5E2+=+0.95\"$ \r
\n" ); document.write( "\n" ); document.write( "The first equation is equivalent to $\"k%2F%282%2Asqrt%28n%29%29+=+0.05\"$ , or $\"sqrt%28n%29+=+10k\"$ .
\n" ); document.write( "==> $\"n+=+100k%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "But the second equation gives $\"k%5E2+=+20\"$ .
\n" ); document.write( "==> n = 100*20 = 2,000 tosses. \n" ); document.write( "

\n" );