# Risk neutral probability for each of 3 states

I need help to find the risk-neutral probability for states 1,2 and 3

I have two stocks: A and B.

The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);

The price of B today is 100 and in a year it will be worth 94(S1), 134(2) or 54(S3)

The annual rf rate is 2%

I did the following for stock A: up : 288/180-1= 0,6 down :120/180-1 =-0,3333

Finding prob for s1: (2- (-33,33))/ 60- (-33,33)) = 0,3785

Then tried to solve for p2 and then p3:
PV= (0,3785* 288 + p2*180+ (0,6215 -p2 )* 120)/1,02= 180

These probabilities don't add up when I try to find the price of stock B, so they are clearly wrong. Can you tell me how to find the right probabilities?

Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation $$p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 = 180$$ is underdetermined, there are infinitely many solutions to it. Now if you solved the equation system \begin{align*} p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 & = 180 \\ p_1 94 + p_2 134 + (1 - p_1 - p_2) 54 & = 100 \end{align*} there you have two unknowns and two equations, so usually this would admit exactly one solution.