Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~
D
(a
0, …, a
d
), Pr(B = (0, …, 0, 1, 0, …, 0)) = a
i
/ a with a = ∑
i=0
d
a
i
, and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution B
k
(a
0, …, a
d
) with k an integer such that the above result holds when B follows B
k
(a
0, …, a
d
) and when Y ~ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a
0 = · · · = a
d
= 1.