How is it possible for the expected value of a game to be losing 65 cents of the only outcomes are win $2 or lose $1

Let x be the probability of winning and let y be the probability of losing. Then we can write the following equation:
2x - y = -0.65 (1)
the sum of the probabilities of winning and losing must equal 1.
x + y = 1(2)
Rearranging (2), gives:
y = 1 - x (3)
Plugging the expression for y from (3) into (1), we get:
2x - 1 + x = -0.65
which simplifies to:
3x = 0.35
Therefore x = 0.117 and y = 0.883.
The expected value of the game will be a loss of 65 cents if the probability of winning is 0.117 and the probability of losing is 0.883.

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Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.