4. 1 win 47.5% of the games I play against the devilish Schnapsen program Doktor Schnaps, and each game is independent of the other games played. Define a "run of bad luck" to be 45 games in a row of which I win a fraction less than 11/30. In 7538 games, what is my expected number of runs of bad luck? These runs are allowed to overlap so, for instance, 47 games in a row can contribute up to 3 runs of bad luck of 45 games each. Give your answer to 4 significant digits. (Hint: Define a Bernoulli random variable for each game played. Then define a random variable X\ to be the average of the next 45 Bernoulli random variables starting at game i. Use the Central Limit Theorem to approximate the probability that Xj is less than 11/30 The numbers have been chosen so that you don't have to worry about applying a continuity correction (unless you do something like deciding that winning at most(45) = 16.5 games is equivalent to winning at most 16 games, in which case you've undone the continuity correction that I built into the problem for you). There was a similar notion of runs of bad luck in Homework 4, and it might be helpful to review how you solved that exercise.) For your interest, in the last 7538 games I have played against Doktor Schnaps, I have had 616 runs of bad luck. 4. 1 win 47.5% of the games I play against the devilish Schnapsen program Doktor Schnaps, and each game is independent of the other games played. Define a "run of bad luck" to be 45 games in a row of which I win a fraction less than 11/30. In 7538 games, what is my expected number of runs of bad luck? These runs are allowed to overlap so, for instance, 47 games in a row can contribute up to 3 runs of bad luck of 45 games each. Give your answer to 4 significant digits. (Hint: Define a Bernoulli random variable for each game played. Then define a random variable X\ to be the average of the next 45 Bernoulli random variables starting at game i. Use the Central Limit Theorem to approximate the probability that Xj is less than 11/30 The numbers have been chosen so that you don't have to worry about applying a continuity correction (unless you do something like deciding that winning at most(45) = 16.5 games is equivalent to winning at most 16 games, in which case you've undone the continuity correction that I built into the problem for you). There was a similar notion of runs of bad luck in Homework 4, and it might be helpful to review how you solved that exercise.) For your interest, in the last 7538 games I have played against Doktor Schnaps, I have had 616 runs of bad luck