Equation (4.10) for covariance indicates exactly what covariance is: the weighted average of products of deviations from

Question:

Equation (4.10) for covariance indicates exactly what covariance is: the weighted average of products of deviations from the means, weighted by the joint probabilities. However, the computing formula for covariance, Equation (4.12), is more convenient for spreadsheet calculations. Show algebraically that the two formulas are equivalent.