Applications - A Guide to MATLAB for Beginners and Experienced Users - page 153

Here we see very clearly the shape of a bell curve. Though we haven't gained
that much in terms of knowing how likely the house is to be behind after 100
games, and how large its net loss is likely to be in that case, we do gain
confidence that our results after 1000 trials are a good depiction of the
distribution of possible outcomes.
Now we consider the net profit after 1000 games. We expect on average
the house to win 510 games and the player(s) to win 490, for a net profit of 20
units. Again we start withjust 100 trials.

hist(profits(1000, 100), -100:10:150); axis tight

15

10

5

0

− 100 − 50 0 50 100 150

Though the range of observed values for the profit after 1000 games is
larger than the range for 100 games, the range of possible values is 10 times
as large, so that relatively speaking the outcomes are closer together than
before. This reflects the theoretical principle (also a consequence of the
Central Limit Theorem ) that the average “spread” of outcomes after a large
number of trials should be proportional to the square root of n , the number of
games played in each trial. This is important for the casino, since if the
spread were proportional to n , then the casino could never be too sure of
making a profit. When w e i ncrease n by a factor of 10, the spread should only

increase by a factor of √ 10, or a little more than 3.
Note that after 1000 games, the house is definitely more likely to be ahead
than behind. However, the chances of being behind are still sizable. Let's
repeat with1000 trials to be more certain of our results.

hist(profits(1000, 1000), -100:10:150); axis tight