Find the expected value (or expectation) of the games...

Find the expected value (or expectation) of the games described.
The game consists of drawing a card from a deck. You win $\$ 100$ if you draw the ace of spades or lose $\$ 1$ if you draw any other card.

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Key Concepts

Expected Value

The expected value is a fundamental concept in probability theory that represents the long-term average outcome of a random variable after many trials. It is calculated by summing the products of each outcome and its corresponding probability, thereby providing a measure of the central tendency of a probability distribution.

Random Variable

A random variable is a numerical representation of the outcomes of a random phenomenon. In probability and statistics, random variables allow us to quantitatively describe the outcomes of experiments or games, and they can be either discrete or continuous.

Probability Distribution

A probability distribution specifies the probabilities of all possible outcomes of a random variable. For discrete random variables, this distribution details how probabilities are assigned to each distinct outcome, enabling the calculation of measures such as expected value.

Payoff Structure

The payoff structure describes how different outcomes of an experiment or game correspond to specific numerical rewards or losses. Understanding the payoff structure is essential when calculating the expected value, as it involves weighting each payoff by its probability of occurring.

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