Hi,
I am quite sure that 4/9 is the right result.
@sameer568: To my mind you are wrong. Indeed there is only one way of getting to the result S1 S1 S1 S1 (i.e. the first secretary is assigned to all four reports), however there are actually 4 ways of getting to the result S1 S1 S1 S2. Even though order doesn't matter for the result, it does matter for calculating the probabilities. Intuitively that comes to light when you consider that the second secretary could be assigned to the 1st, 2nd, 3rd or 4th report.
--> When you flip a coin 2 times and you are only interested in the overall result (H = Head, T = Tail), i.e. either TT or HH or HT, you won't assign a probability of 1/3rd for each of the events, but rather 1/4 respectively for each event, namely TT, HH, HT and TH, however, ultimately HT equals TH, so that this event has the probability of 1/2.
In the secretary case, in my mind you should approach the problem using the
multinomial coefficient.
(E.g. English Wikipedia:
https://en.wikipedia.org/wiki/Multinomial_theorem
or German:
https://de.wikipedia.org/wiki/Multinomialkoeffizient).
So what you do is: Total number of assignment orders are: 3^4 = 81, because every time a report is assigned (4 times), there are 3 secretaries which can be chosen.
Now we are searching for the possibility that one of the secretaries gets assigned to 2 reports, the other two to 1 report respectively. I.e. we want to know how many alternatives there are to split a set of the cardinality 4 (number of reports) into three sets of the cardinality 2, 1, and 1 (numbers of assigned reports per secretary) respectively. Using the concept of the multinomial coefficient (actually an expansion of the binomialcoefficient concept) you get the following result.
First, there are 3 alternatives of splitting the set of cardinality 3 into 2 subsets of the cardinalities 2 and 1, namely one secretary with 2 reports (the set with cardinality 1 because it contains 1 secretary) - and two secretaries with 1 report each, comprised by the set with cardinality 2 (because it contains 2 secretaries). The three alternatives are: (2,1,1) , (1,2,1) and (1,1,2), which can also be calculated using the multinomial coefficient 3C(2, 1) = 3!/(2!*1!) = 3
Now we calculate the number of alternatives for one of these events as a result, e.g. for (2, 1, 1):
--> There are 12 alternatives of splitting a set of cardinality 4 into 3 subsets of the cardinalities 2, 1 and 1 respectively, so that the first secretary gets 2 reports, the other secretaries 1 each (i.e. event (2,1,1)):
--> 4C(2, 1, 1) = 4!/(2!*1!*1!) = 12
(Intuition: By dividing by 2!*1!*1! you correct for the number of permutations within the subsets, as these are irrelevant.)
There are also 12 alternatives respectively for the events (1,2,1) and (1,1,2). So there are 36 alternatives of 81 that fulfill the condition that each secretary is assigned at least 1 report.
The possibility for the event is thus: 36/81 =
4/9
It's hard to explain that concept this way, but I think it is right. Important to understand is that even though order doesn't matter for the result it matters for calculating the probabilities!
