TITLE: A wavy histogram from a maximum likelihood algorithm
QUESTION [10 upvotes]: This question is based on a Puzzling Stack question I answered.
Suppose you have a loaded $10$-sided die that gives one value with probability $\frac7{25}$ and the rest with probability $\frac2{25}$ each, but you do not know the favoured value. A maximum likelihood method to determine this favoured value is as follows:

Set probabilities $p_1=\dots=p_{10}=\frac1{10}$, where $p_i$ is the likelihood of $i$ being the favoured value.
Roll the die repeatedly; for each value $c$ that comes up multiply $p_c$ by $3.5$, the ratio of probabilities of a favoured face to an unfavoured face. Normalise after each update and stop when one $p_i$ becomes dominant enough.

This strategy is quite efficient, for simulating it $4121000$ times gave a mean of about $35.47$ rolls until one $p_i$ exceeded $0.99$ and a median of $32$ rolls. The histogram of rolls needed, however, looks very odd with its wavy top:

Is there any logical explanation for the histogram's peculiar shape? The number of simulations I made certainly seems high enough to rule out sample size as an explanation.

Here is the histogram corresponding to a d3 with one value twice as likely ($1/2$) to come up as the other two ($1/4$):

Clearly there is a dependence on the number of possibilities, but what kind?

REPLY [3 votes]: To corroborate Oscar Cunningham’s comment and HighDiceRoller’s answer, here’s a histogram disaggregated according to the number of times the favoured face came up. The aggregated histogram is the purple one. You can see nicely how it’s composed of the individual components whose spread and overlap increases with the number of favoured rolls; at low values they’re sufficiently separated to make their peaks stand out in the sum.