A fair coin will be tossed 200,000 times. Let X denote the number of Tails. (a) What is the expected value and the standard deviation of X? (b) Consider a game in which you have to pay $5 in order to earn $log10(X) when X > 0. Is this a fair game? If not, your expected profit is positive or negative?

Answer and explanation:Given : A fair coin will be tossed 200,000 times. Let X denote the number of Tails.To find : (a) What is the expected value and the standard deviation of X? Given that total number of tosses is n=200000 Probability of getting tail in single toss is p=0.5 Expected value is given by, [tex]E(X)=n\times p[/tex][tex]E(X)=200000\times 0.5[/tex][tex]E(X)=100000[/tex]Standard deviation is given by,[tex]SD=\sqrt{n\times p\times q}[/tex][tex]SD=\sqrt{200000\times 0.5\times 0.5}[/tex][tex]SD=\sqrt{50000}[/tex][tex]SD=223.60[/tex](b) Consider a game in which you have to pay $5 in order to earn [tex]\$\log_{10}(X)[/tex] when X > 0. Is this a fair game? If not, your expected profit is positive or negative?We have to pay $5 to get [tex]\$\log_{10}(X)[/tex]Minimum number of tails required to get $5 is 100000 . Since, we get X=100000 with Probability 0.5 and for winning we need more number of tosses.Probability of losing is more than profit hence it's biased test . As expected number of tails =100000 So profit is given by,[tex]P=\$\log_{10}(100000)-5[/tex][tex]P=\$\log_{10}(10^5)-5[/tex][tex]P=5-5[/tex][tex]P=0[/tex]Therefore, The profit is zero.