# How to find nash equilibrium of a mixed game?

So I have this question with this solution. I have no problem calculating the pure strategy equilibrium given by (3,3) and (1,1).

However, this is how I would calculate the mixed strategy as taught by by old professor:

If player 1 is on the rows, and player two is on the columns, and the rows represent UP and DOWN while the columns represent LEFT and RIGHT.

Then UP is represented by P and DOWN is represented by 1-p.

Then LEFT is represented by Q and RIGHT is represented by 1-q.

The expected payoff of Player 1 will be:

Expected payoff of UP= 3q -1(1-q)= 3q-(1-q)

Expected payoff of DOWN= 3q+1(1-q)= 3q+(1-q)

If i equate this two equations then I get that q=1 and consequently 1-p=0.

The expected payoff of Player 2 will be:

Expected payoff of LEFT= 3p-1(1-p)

Expected payoff of RIGHT= 3p+(1-p)

If i equate this two equations then I get that p=1 and 1-p=0..

Now how come the answer sheet gives different results?

Shouldn't the nash equilibria be (1,0), (1,0)?

Where is this coming from ? Would I be right if i stated that the nash equilibria is (1,0), (1,0)?