I am brand new to the site.  Here is Henry's plot using Ian's formula.  In Excel, I did BINOM.DIST(i,4,0.333,TRUE) for i = 0 to 4.  Then, I did 1-BETA.DIST(0.333,1+i,4-i,TRUE) for i = 0 to 3.99.  I have used this equivalence before without understanding why it works.  The cumulative beta distribution is an integral for p = 0 to x.  The binomial distribution is a sum for k = 0 to i.   

[![Plot][1]][1]


  [1]: https://i.sstatic.net/NfdHh.png