There has always been some confusion about what the probabilities in Appendix E of the USGA Handicap System actually represent. John Paul Newport of the Wall Street Journal, for example, wrote a player with a handicap between 13 and 21 will only play 3 strokes better than his handicap once in 43 rounds. Newport cites the website of Dean Knuth, former Senior Director of Handicapping at the USGA for his probability estimate. Knuth cited the work of F.P. Engel who defined the Net Differential as:
 Knuth, D.L., F.J. Scheid, and F.P Engel, “Outlier identification procedure for reduction in handicap,” Science and Golf II, E & F Spon, London, 1994, pp. 228-233.
Net Differential = (Adj. Score – Handicap) - Course Rating
It appears Knuth used Engel's probabilities for a player with a 13-21 handicap. These probabilities were for beating your Handicap by N strokes or better.
Some time later, the USGA changed the definition of Net Differential to mean beating your Index, but the probabilities remained the same. This change was made because either the USGA thought it expedient or did not understand that “beating your Index” and “beating your Handicap” were not the same thing. The USGA was informed of the problem, but chose to ignore it.
To demonstrate the difference, a player’s adjusted score under the two definitions of Net Differential is computed below:
Beat Your Handicap
1) Net Differential = (Adj. Score – Course Rating) - Handicap = -N
2) Adj. Score = -N + Course Rating + Handicap
Beat Your Index
3) Net Differential = (Adj. Score – Course Rating)·(113/Slope Rating) – Index = -N
4) (Adj. Score-Course Rating)·(113/Slope Rating) - Handicap·(113/Slope Rating) = -N
5) Adj. Score = -N·(Slope Rating/113) + Course Rating + Handicap
Equations 2) and 5) show it takes the same score to beat your Handicap or Index (i.e., N = 0). For other values of N, however, a player needs a lower score to beat his Index than he does to beat his Handicap if the Slope Rating is greater than 113. For example, if the Slope Rating is 150 and N= 6, a player would have to score approximately 2 strokes lower to beat his Index by -6 than he would to beat his Handicap by -6.