# Drawing a Probability simplex

There are 3 possible payoffs - \$4, \$9 and \$36. The utility function for these payoffs is $$\sqrt x$$. I have to find all the lotteries preferable over getting$9 with probability 1 in a probability simplex. The answer is the shaded area in the probability simplex below.

I am not able to find the 3D equation of line required for it. But I have approached the problem as follows:

Any point on line BC gives expected utility of $$6p_{3} + 3(1-p_{3}) = 3p_{3} + 3 > 3$$, so all points on BC are to be included.

All points on AB are eliminated as expected utility on them is less than 3.

Only those points on AC are included which give the expected utility greater than 3 i.e.

$$2p_{1} + 6(1-p_{1}) = 6 - 4p_{1} >3 \implies p_{1} < \dfrac{3}{4}$$

From here how do we arrive at point D ?

In your last inequality, point $$A$$ is where $$p_1=1$$ and point $$C$$ is where $$p_1=0$$. So the condition $$p_1<\frac34$$ is captured by the line segment $$\overline{CD}$$.