Expected Value of a Game

Calculate the expected value of winning a game where if she wins, she receives $500, and if she loses, she loses$100. The probability of winning the game is 20%.

In this problem, you're asked to calculate the expected value of a financial game, which is a common application in probability theory. Expected value is a crucial concept that provides insight into the potential outcomes of probabilistic events, particularly in the context of random variables and gambling scenarios. When you calculate the expected value, you're essentially finding the mean of all possible outcomes, weighted by their probabilities. This allows you to determine whether participating in a game is statistically advantageous or not.

The process involves multiplying each outcome by the probability of its occurrence and then summing these values. For this problem, you'll need to factor in both winning and losing scenarios. The key here is to carefully consider the gains from winning and the losses from losing, which will help you compute the overall expected value. By understanding this concept and how to apply it, you not only gain insight into the problem itself but also into larger applications, such as risk assessments and decision-making processes in uncertain environments.

Additionally, expected value plays a vital role in various fields ranging from finance to actuarial science, where understanding the long-term averages of random outcomes is essential. Mastery of this concept also lays the groundwork for more advanced statistical topics such as variance and standard deviation, which measure the spread of possible outcomes around this expected value.