Find the expected value of the winnings from a game that has the following
payout probability distribution:
Payout ($)| 0 4 6 8 10
Probability 0.45 0.3 0.1 0.1 0.05
Expected Value = [?]
Round to the nearest hundredth.

2.65

Step-by-step explanation:

Multiply each payout by its probability, then add those products.

See the attached image.

The first column has the payouts.  The second column has the probabilities. The third column has the results of multiplying a payout by its probability.

The sum of the entries in the third column is 2.65

1 number

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.