3-12 . Find the expected value (or expectation) of the games...

$3-12$ . 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

Sample Space

The sample space is the complete set of all possible outcomes of a random experiment. Understanding the sample space is crucial because it provides the context and boundaries within which probabilities are assigned, thereby affecting the calculation of expected values.

Discrete Random Variable

A discrete random variable is one that can take on only specific, separate values. Each of these values has an associated probability, and the concept is essential in determining outcomes and their probabilities for the purpose of computing expectations.

Probability

Probability is a measure that quantifies the likelihood of an event occurring, typically expressed as a number between 0 and 1. It underpins the calculation of expected values by assigning weights to the outcomes of a random experiment.

Expected Value

Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event if it were repeated many times. It is calculated by multiplying each possible outcome by its probability and then summing all these products, providing a measure of the central tendency of a probability distribution.

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