I think I finally sorted out what chance performance is on the 8-arm.
The answer - 36.36%.
Other interesting facts that I had to compute along the way -
Probability of getting 100% by chance: 0.3%
Probability of getting 75% by chance: 8.4%
Probability of getting 50% by chance: 38.18%
Probability of getting 25% by chance: 42.4%
Probability of getting 0% by chance: 10.6%
Attached is a photograph showing my work.
Computing these values uses the following basic approach:
First, for each of the possible score levels (100%, 75%, 50%, 25%, and 0% in the case of showing four arms at study and then scoring the first four arms entered at test), compute the probability of getting that score by chance.
For a given score level, this looks something like this:
C(4,score*4) * CR(4+(4*score),4*(1-score))
CR(8,4)
C(n,r) indicates the number of combinations that exist from drawing r items from a set of n with NO replacement. There are many calculators available online to compute this. I used this site.
This part of the equation is used to represent the number of ways the rat could get this many correct given the total set of correct arms available.
CR(n,r) indicates the number of combinations that exist from drawing r items from a set of n WITH replacement. I used this site to compute this.
When used in the numerator, this part of the equation is used to represent the number of ways the rat could get an error. Errors can be generated by going into any of the arms shown at study (assuming this is a win-shift paradigm) or by entering any arm that has already been entered. This is why the total number of n is higher when performing this calculation for higher score levels - with the larger number of correct arms already entered, there are more opportunities for error.
When this is used in the denominator, it represents the total number of combinations of arms the rat could make in the first four entries given the total available number of 8 arms regardless of whether they are accurate or not.
Score is the accuracy as a decimal - 100% is 1.0, 25% is 0.25.
For example, the probability of getting 25% correct by chance can be computed like this:
C(4 , (0.25*4)) * CR(4+(4*0.25) , 4*(1-0.25))
CR(8,4)
or
C(4 ,1) * CR(4+1 , 4*0.75)
CR(8,4)
or
C(4,1) * CR(5,3)
CR(8,4)
Putting these values into the calculators linked above gives:
4 * 35
330
or
42.4% (as noted above)
Once you've computed the probabilities of getting each of the different scores, take the weighted mean of all of the scores to determine the expected score:
For example, using the numbers from above:
(100% * 0.03) + (75% * 0.084) + (50% * 0.382) + (25% * 0.424) + (0% * 0.106) = 36.36%
Note - all of this assumes zero memory on the part of the rat. That is, it assumes that errors are just as likely to come from re-entering an arm as they are to come from entering one of the study arms.
If we do the process all over again and assume perfect working memory on the part of the rat but no memory of the study phase, the result is different.
In the case of intact working memory, chance is 50%.
Probability of getting 100% by chance: 1.43%
Probability of getting 75% by chance: 22.86%
Probability of getting 50% by chance: 51.43%
Probability of getting 25% by chance: 22.86%
Probability of getting 0% by chance: 1.43%
To come by these values I used the following approach:
C(4,score*4) * C(4,4*(1-score))
C(8,4)
For example, for score = 25%
C(4,1) * C(4,3)
C(8,4)
or
4 * 4
70
or
22.86%
Notice that this does not use replacement because it assumes the rat will not enter an arm a second time in a single test phase given intact working memory.