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use chebyshev's theorem and theorem 5.3 to verify that the probability is at least 35/36 that
a. in 900flips of a balanced coin the proportion of heads will be between 0.40 and 0.60
b. in 10000flips of a balanced coin the proportion of heads will be between 0.47 and 0.53
c. in 1000000 flips of a balanced coin the proportion of heads will be between 0.497and 0.503
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chebyshev's theorem：若X為隨機變數，平均數為μ，標準差為σPro（｜X-μ｜≦kσ）≧1-1/k^2即Pro（｜X-μ｜/σ≦k）≧1-1/k^2 a. in 900flips of a balanced coin the proportion of heads will be between 0.40 and 0.60E(Phat)=p=0.5，σ（Phat）=σ(P)/√n=√（0.5*0.5）/√900=1/60Pro(0.4≦Phat≦0.6)=Pro(|Phat-0.5|≦0.1)= Pro(|Phat-0.5|/(1/60) ≦6) ≧1-1/6^2=35/36
b. in 10000flips of a balanced coin the proportion of heads will be between 0.47 and 0.53 E(Phat)=p=0.5，σ（Phat）=σ(P)/√n=√（0.5*0.5）/√10000=1/200Pro(0.47≦Phat≦0.53)=Pro(|Phat-0.5|≦0.03)= Pro(|Phat-0.5|/(1/200) ≦6) ≧1-1/6^2=35/36
c. in 1000000 flips of a balanced coin the proportion of heads will be between 0.497and 0.503E(Phat)=p=0.5，σ（Phat）=σ(P)/√n=√（0.5*0.5）/√1000000=1/2000Pro(0.497≦Phat≦0.503)=Pro(|Phat-0.5|≦0.003)= Pro(|Phat-0.5|/(1/2000) ≦6) ≧1-1/6^2=35/36