Batting averages. Suppose that a baseball player's long-run batting average (number of hits per time at bat) is 300. Assuming that each time at bat yields a hit with a consisted probability, independently of other times, what is the chance that the player's average over the next 100 times at bat will be

a) 310 or better?

b) .330 or better?

c)  .270 or worse?

d) Suppose the player tends to have periods of good form and periods of bad form would different times at bat then be independent? Would that tend to increase or decreases the above chances?

e) Suppose the player actually hits .330 over the 100 times at bat. Would you be convinced that his form had improved significantly? Or could the improvement just as well be due to chance?