"If a fair die is rolled 200 times, count the number of 1’s. Give an upper bound for the probability that the count of 1’s stays below 8. " So here for Markov inequality, P(X>=8)<=E[X]/8. So here, E[X]=200*(1/6). So, P(X>=8) <=200/(8*6) <= 4.1666. Markov gives lower bound. So for upper bound, 1-4.166= -3.166 Is the answer correct? How can the probability get less than 0?
2026-04-01 02:52:46.1775011966
Markov Inequality question confusion bound problem
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$P(X\le 8)=P(Y\ge 192)\le E[Y]/192$
Now, $E[Y]=200\times 5/6=166.666$ and the above inequality gives the estimate $$ P(X\le 8)\le 0.86805... $$