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Probability is the likelihood or chance of a specific event occurring. In our scenario, we want to know the odds of choosing a finger trap from a box full of various party favors. To do this, we need to use the probability formula. The formula to calculate probability is: \[ P( ext{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

To apply this to our problem:

• The total number of outcomes refers to all party favors combined in the box. After summing them up, it's clear there are 42 items in total (12 hats, 15 noisemakers, 10 finger traps, and 5 bags of confetti).
• The number of favorable outcomes is the number of finger traps (the event we are interested in), which is 10.

This gives us the probability of pulling out a finger trap as:\[ P(F) = \frac{10}{42} \]

Probability is a useful concept for predicting outcomes and making decisions based on available data.

Understanding what constitutes a favorable outcome is key to calculating probability. In the context of our party favors problem, a favorable outcome is defined as the event of selecting a finger trap from the box.

Here's how we determine and count favorable outcomes:

• We look at all the different types of items in the box (hats, noisemakers, finger traps, confetti bags) and identify the specific event we are interested in.
• In this exercise, we are interested in just one type: finger traps. Thus, the number of favorable outcomes is the number of finger traps, which totals 10.

Your outcome of interest could be anything else depending on your goal — for example, choosing a hat would represent a different favorable outcome. In probability exercises, clearly identifying the event you are interested in is fundamental.

This understanding allows you to correctly plug the numbers into the probability formula and make accurate predictions.

In many probability problems, you end up with a fraction that represents the probability of an event occurring. Simplifying fractions not only makes the number easier to interpret but also ensures clarity in your final answer.

After calculating the fraction for our probability exercise, we end up with:\[ \frac{10}{42} \]

To simplify this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. Here’s the step-by-step:

• The GCD of 10 and 42 is 2. We divide both the numerator (10) and the denominator (42) by this number to reduce the fraction.
• This simplifies our fraction to: \[ \frac{10 \div 2}{42 \div 2} = \frac{5}{21} \]

The fraction \( \frac{5}{21} \) is now in its simplest form. Simplifying fractions is an essential skill in probability as it gives you a cleaner and more digestible result, and it is especially useful if you need to compare probabilities or carry out additional calculations.