Solution
Monochromatic Triangles
Show the problem again
Six points are placed in general position and all 15 segments between pairs of them are drawn. Each segment is independently coloured red or blue with probability 1/2 each. A triangle (three of the points together with the three segments joining them) is monochromatic if its three segments all have the same colour. What is the expected number of monochromatic triangles? Give an exact number.
Worked solution
The answer is 5. There are C(6,3) = 20 triangles. A given triangle is monochromatic when its three edges are all red or all blue, which has probability 2 × (1/2)³ = 1/4. By linearity, the expected count is 20 × 1/4 = 5. For context, Ramsey theory (R(3,3) = 6) guarantees at least one monochromatic triangle in every colouring, and Goodman's theorem sharpens this to at least two; the random colouring does considerably better on average.
Source: Classic indicator-variable exercise related to Ramsey theory; no single originator. Statement written for AxiomIQ.