One possible method for solving Problem B2 is to use two 10-sided dice, one red and one blue. The sub-region can then be determined by the two dice (in the order red, and then blue).
You might notice that the random selection process will sometimes produce duplicates. There is a greater than one-third chance that 10 numbers picked at random between 00 and 99 will produce at least one duplicate, and almost a 90% chance that 20 such numbers will produce at least one duplicate.
For instance, you might find that seven tosses of the dice produced these sub-region choices:
19 22 39 50 34 05 39
If we do not want duplicates, we can skip them until we get 10 distinct numbers, for example:
19 22 39 50 34 05 75 62 87 13
This is called sampling without replacement, since each time we choose a sub-region we remove it from the list of sub-regions we can choose on the next toss of the dice. In some experiments, it may be impractical or impossible to exclude duplicates from the random selection process. If duplicates are allowed, it is called sampling with replacement.
The 10 distinct numbers (19, 22, 39, 50, 34, 05, 75, 62, 87, 13) correspond to these 10 sub-regions:
Here is a look at the number of penguins in each of the 10 sub-regions we selected:
The estimate of the total number of penguins for the entire region based on this random sample of 10 sub-regions is as follows:
100 x [(5 + 6 + 6 + 7 + 5 + 2 + 1+ 5 + 5 + 3)/10] = 100 x (45/10) = 450