I'm looking for a simple, symmetric multivariate distribution for $N$ Bernoulli variables with the following properties:

Each individual variable takes on values 1 or 0

Fix a subset of $M$ variables. Let $P_K$ be the probability that conditional on $K$ of these $M$ variables taking on value 1, the $N$th variable is 1. Want: $P_K$ increasing in $K$.

A parameterized functional form with a single parameter, so that at the extremes, we arrive at the case of perfect correlation and at independence.

Hopefully this is not too simple a question to ask on mathoverflow, but I've had a tough time coming up with a simple distribution that does this. Many thanks.