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In probability, an 'event' is essentially any collection of possible outcomes. When we talk about the probability of an event, we are referring to the chance that this specific event will occur out of all the possible outcomes.

To calculate this, we use the formula for event probability, which is given by:

• \[ P(Event) = \frac{\text{Number of favorable outcomes for the event}}{\text{Total number of possible outcomes}} \]

For example, in our box of party favors, the event might be selecting a hat. In this case, you count how many hats there are (which are your favorable outcomes), and divide it by the total number of party favors. This gives you the probability of the event occurring.

Fractions are a mathematical expression representing division or part of a whole. Often fractions can be simplified for easier understanding and interpretation.

Simplifying a fraction means finding an equivalent fraction where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. To do this, locate the greatest common divisor (GCD) of the numerator and the denominator and divide both by this GCD.

For example, if you have a fraction \(\frac{12}{42}\), find the GCD of 12 and 42, which is 6 in this case. Then, divide both the numerator and denominator by 6:

• \[\frac{12 \div 6}{42 \div 6} = \frac{2}{7} \]

The fraction is now simplified, and it's easier to interpret in terms of probability or other contexts.

In any probability experiment, knowing the total number of possible outcomes is crucial. It allows you to understand the base against which all probabilities are measured. Total outcomes can be counted by simply adding up all the different distinct items involved.

For example, in our box of party favors, you have several kinds of items: hats, noisemakers, finger traps, and confetti bags. To count total outcomes, you just add:

• \(12 + 15 + 10 + 5 = 42\)

This sum, 42, represents the total number of party favors you could possibly pick if you randomly selected one from the box.

Knowing the total outcomes allows us to calculate the probability of each specific event, such as picking a hat, by comparing the number of favorable outcomes to this total.

There are various steps to calculating the probability of an event, but the process is straightforward. First, determine the event of interest and its outcomes, then count these favorable outcomes.

Next, calculate the total number of possible outcomes by adding all items together, as we did before. Once both numbers are known:

• Plug them into the probability formula: \[ P(Event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Use this formula to find the initial probability, then simplify the resulting fraction if necessary. In our example, calculating the probability for picking a hat:

• Number of hats = 12
• Total outcomes = 42

So the probability formula becomes: \[ P(Hat) = \frac{12}{42} \]

Simplify to get \[ \frac{2}{7} \].

And that's the step-by-step guide to probability calculation.