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Probability is a concept that measures the likelihood of an event occurring. It ranges from 0 to 1, where 0 means the event will not happen, and 1 means it will happen for sure.
For instance, in the spinner game, each outcome like winning \(2, \)5, or $10 has its own probability. These probabilities tell us how likely each outcome is when you spin the spinner.
The probabilities for the outcomes in the table are \(\frac{1}{3}\), \(\frac{1}{2}\), and \(\frac{1}{6}\). When combined, they must add up to 1 to show a complete set of possible outcomes. This ensures we account for every possible result of spinning the spinner.

An outcome is the result of a random event. In our context, the "outcome" is what you get when you spin the game wheel.
Each slice of the spinner is an outcome with a specific monetary win, like $2, $5, or $10. The actual winnings depend on where the spinner arrow lands.
Knowing the possible outcomes is crucial for calculating the expected value because they determine the rewards associated with each spin.
It's important to remember that even unlikely outcomes still affect the total expected value. Probabilities accompanying each outcome predict long-term averages when the game is played many times.

A spinner game involves a circular wheel divided into segments, each representing a different outcome with its own probability. It's a fun way to understand probability in a visually engaging format.
In our exercise, the spinner is labeled with monetary outcomes: $2, $5, and $10, each segment with an associated probability.
As you spin the wheel, the arrow will eventually point to one segment, determining the outcome. The randomness of each spin makes spinner games an excellent example to study mathematical expectation and probability models.
It represents real-world uncertainty and helps in learning how to calculate expected outcomes effectively over numerous attempts.

Mathematical expectation, often referred to as expected value, is an average value someone can expect over many trials of a random activity.
Calculating it involves multiplying each outcome by its probability and adding these products together. This gives us an idea of what we might "expect" to win or lose on average after spinning the wheel many times.
Using our exercise, we multiply each monetary value by its probability: \(2 by \(\frac{1}{3}\), \)5 by \(\frac{1}{2}\), and $10 by \(\frac{1}{6}\). Adding up these values, we find that the expected value of the game is \(\frac{29}{6}\).
Understanding mathematical expectation helps us make better decisions in situations involving chance. It can reveal whether a game is favorable or not based on expected returns.