Dice probability

[GDC]

You ve just been hired as a theft prevention officer at one of Las Vegas's casinos. You've been given a pair of dice and have been asked to determine whether the number '7' is appearing too many times. You roll the dice 100 times and note that '7' shows 23 times. Using the "Range Rule of Thumb" can this result be reasonably expected from fair dice?

[Adrian]

Out of 36 possible outcomes (11,12,21,13,31,22,...,56,65,66),
6 outcomes will result in a sum of the number '7' (16,61,34,43,25,52)
The number '7' appears most oftern in all the numbers '2' through '12'
Thus P('7' appears)=6/36=0.1667
To use the Range Rule of Thumb, we must find the probability of the number appearing least often.
Only the combination 11 results in the number '2'.
P('2'appears)=1/36

range=6/36-1/36=5/36

From the Range Rule of Thumb, standard deviation is roughly 1/4 of the range.
sd=(1/4)*(5/36)=0.03472

In our experiment, '7'appeared 23 out of 100 times, p=0.23
0.23-0.1667=0.0633
This is 0.0633/0.03472=1.82 standard deviation away from the theoretical value.
This is less than 2 sd away, we consider is not too far from the thretical value.
This result can be reasonably expected from fair dice.