The Encyclopedia of Beekeeping (Morse & Hooper 1885) suggest a sample of 30
bees for disease diagnosis (in those days it was only acarine, nosema and
amoeba that we had to worry about - apart from the foulbroods), giving the
reason for the sample size as the following table of probabilities:

Percentage of infected       Number of bees to be examined to give the
indicated
bees in colony                   percentage chance of finding at least one
diseased bee

                                        80%        90%        99%

90                                    1             2             2
80                                    1             2             3
70                                    2             3             4
60                                    2             4             5
50                                    3             5             7
40                                    3             6             9
30                                    5             8            13
20                                    7            13           21
15                                   10           18           28
10                                   15           28           44
5                                     31           58           89
1                                    161        298         457

(Hope this table formats OK!)

So by taking a sample of 30 bees, we have (approximately) a 95% chance of
finding a diseased bee if 10% of the colony is diseased or an 80% chance if
only 5% is diseased.  Using the table, you can decide on the sample size
necessary to give you the level of confidence that you require.

Peter Edwards