X, Y and Z are standard normal with every pairwise correlation equal to 1/2. What is the probability that all three are positive?

X, Y and Z are standard normal with every pairwise correlation equal to 1/2. What is the probability that all three are positive?

Approach: Use the orthant formula that writes the trivariate positive probability as 1/8 plus a sum of arcsine terms, one per pairwise correlation, then check the answer against a common factor construction.

1/4. For standard normals the orthant probability is P(X>0, Y>0, Z>0) = 1/8 + (1/(4 pi))(arcsin r_xy + arcsin r_xz + arcsin r_yz). With equicorrelated normals at 1/2, each arcsine equals pi/6, so the value is 1/8 + (1/(4 pi))(pi/2) = 1/8 + 1/8 = 1/4. A construction makes it concrete: three variables equicorrelated at 1/2 can be written as (W + V_i)/sqrt(2) with W and the V_i independent standard normals, and the shared factor W pushes all three the same way, which is why 1/4 exceeds the independent value of 1/8. The two-dimensional arcsine formula 1/4 + arcsin(r)/(2 pi) gives 1/3 at r = 1/2.

Follow-up: What is the probability that all four are positive when four standard normals share a correlation of 1/2?

Key concepts: orthant probability, arcsine formula, equicorrelated normals, shared factor.