Each record in the crosswalk describes a spatial intersection (a shared area of overlap) between a 2000 PUMA and 2010 PUMA. If a 2000 PUMA is identical to a 2010 PUMA, there will be just one record describing their total area. If a 2000 PUMA was split into two 2010 PUMAs, there will be two records for the 2000 PUMA, one for each of its intersections with 2010 PUMAs. The “area of intersection” referred to in the file’s “data_dictionary” sheet is the shared area between each specified 2000 and 2010 PUMA.
Given your scenario, there are two strategies that would be appropriate, depending on your exact needs.
First, there is a “probabilistic” approach, as you suggest. For your settting, the most suitable probability to use is given in the “pPUMA10_Pop10” column. This identifies the “estimated percent of the 2010 PUMA’s 2010 population that lies in the area of intersection”, i.e., how much of the 2010 PUMA’s population resided in this 2000 PUMA? If 80% of a 2010 PUMA’s population resided in a particular 2000 PUMA, then you could assign each resident of the 2010 PUMA a 80% probability of having lived in that 2000 PUMA.
Alternatively, to reduce uncertainty and achieve greater accuracy, you could aggregate all of your data up to larger “ConsPUMAs”, which are “aggregations of one or more 2010 PUMAs that, in combination, align closely with a corresponding set of 2000 PUMAs”. See the ConsPUMA Definitions page for more details. The crosswalk file indicates which ConsPUMA each 2000 and 2010 PUMA is in, and there are also two files on the ConsPUMA Definitions page that identify the sets of 2010 PUMAs and 2000 PUMAs comprising each ConsPUMA. The advantage of ConsPUMAs would be that there would be no probabilistic assignment–you’d be able to identify (almost) exactly the correct set of residents for each ConsPUMA using either 2000 or 2010 PUMA codes. The disadvantage is that, in areas where there are many discrepancies between 2000 and 2010 PUMAs, ConsPUMAs can be very large. Through the Definitions page, you can find online maps that show the extents of ConsPUMAs relative to PUMAs. You could view those to determine if the areas would be satisfactory units for your analysis.