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

Find the expected value (or expectation) of the games described.
A card is drawn from a deck. You win $\$ 104$ if the card is an ace, $\$ 26$ if it is a face card, and $\$ 13$ if it is the eight of clubs.

Key Concepts

Expected Value

In probability theory, the expected value is a fundamental concept that represents the long-run average outcome of a random process over many trials. It is calculated by multiplying each possible outcome by its probability and summing the products. This concept is used to predict the theoretical average gain or loss over a large number of repetitions of the game or experiment.

Probability of Events

Understanding the probability of events is crucial when determining the outcomes of random processes. In scenarios involving random draws, such as selecting a card from a deck, each event (e.g., drawing a specific card or category of cards) has an associated probability which quantifies how likely that event is to occur. These probabilities form the basis for calculating the expected value.

Random Variable

A random variable is a numerical representation of the outcomes of a random process. In the context of games of chance, it assigns a monetary value or payoff to each outcome. The random variable helps in formalizing the calculation of expected value by allowing the association of monetary rewards with the underlying events dictated by chance.

Finite Sample Space

A finite sample space refers to a set of all possible outcomes in an experiment, which is limited and countable. For instance, a deck of cards constitutes a finite sample space because it contains a fixed number of cards. This concept is important because it allows the precise computation of probabilities for each outcome, which are essential for determining the expected value.

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