6. Answer these questions please

1. In the Chinese Appetizer problem n people are eating n different appetizers arranged on a circular, rotating table. Someone spins the tray so that each person receives a random appetizer. What is the probability that everyone gets the same appetizer as before ? How does this compare with the bound obtained using Markov's inequality ? 2. Consider two games. In game A, each time we play we win $2 with probability 2/3 and lose E1 with probability /3. In game B, each time we play we win (1002 with probability 2/3 and lose 62001 with probability 1/3. What is the expected winnings in both games ? What is the variance ? Using Chebyshev's inequality, compute an upper bound on the probability that after playing 10 rounds of each game the winnings deviate by more that $10 from the expected value. Write a Matlab simulation to estimate the probability that make a loss after 10 rounds of each game. 3. In a poll n turkeys selected independently at random are asked whether they vote for Christmas or not. Let X be the number of yes votes in our sample and use S=X as our estimate of the actual fraction of turkeys who like Christmas. Let Let X:=1 when the ith turkey likes Christmas and 0 otherwise and assume X:~Ber(p). Using Chebyshev's inequality, how big should n be to ensure that this estimate is within 0.04 of the true fraction at least 95% of the time ?1. In the Chinese Appetizer problem n people are eating n different appetizers arranged on a circular, rotating table. Someone spins the tray so that each person receives a random appetizer. What is the probability that everyone gets the same appetizer as before ? How does this compare with the bound obtained using Markov's inequality ? 2. Consider two games. In game A, each time we play we win 62 with probability 2/3 and lose E1 with probability /3. In game B, each time we play we win (1002 with probability 2/3 and lose 62001 with probability 1/3. What is the expected winnings in both games ? What is the variance ? Using Chebyshev's inequality, compute an upper bound on the probability that after playing 10 rounds of each game the winnings deviate by more that +10 from the expected value. Write a Matlab simulation to estimate the probability that make a loss after 10 rounds of each game. 3. In a poll n turkeys selected independently at random are asked whether they vote for Christmas or not. Let X be the number of yes votes in our sample and use S=X as our estimate of the actual fraction of turkeys who like Christmas. Let Let X=1 when the ith turkey likes Christmas and 0 otherwise and assume X-Ber(p). Using Chebyshev's inequality, how big should n be to ensure that this estimate is within 0.04 of the true fraction at least 95% of the time