I wanto construct a probability distribution on three binary valued ($yes, no$) random variables (say $P(X_1, X_2, X_3)$);
The independencies that P has to hold are $X_1 \perp X_2, X_1 \perp X_3, X_3 \perp X_2$ but it should not follow $X_1,X_2 \perp X_3$;
I am successful in preserving the first three dependencies by assigning $P(X_i = yes) = 0.5$ for $1 \le i \le 3$
Using those values, I can calculate $P(X_i,X_j) = 0.25$ for $i \ne j$ and $P(X_1, X_2, X_3) = 0.125$, which satisfies unwanted independence.
Now, the first three properties hold. But it automatically holds unwanted independence $X_1, X_2 \perp X_3$ also.
How to develop distribution that satisfies pairwise independecies but not unwanted one?
Let $X_1,X_2$ be i.i.d. random variables following a Bernoulli distribution with parameter $1/2$, and let $X_3$ be their $\textsf{XOR}$: $$ X_3 \stackrel{\rm def}{=} X_1\oplus X_2 = X_1+X_2\bmod 2\,. $$ Then it is easy to see that $X_i\perp X_j$ for every $i\neq j$ (pairwise independence), that $X_1,X_2,X_3$ are identically distributed (all Bernoulli $1/2$); but of course $X_1,X_2,X_3$ are not jointly independent.