Theory Of Probability

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Rasmussen College, Florida *

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Statistics

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May 29, 2024

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Module 03 Theory of Probability  Theory of Probability Introduction to the Theory of Probability With the possible exception of death and taxes, pretty much everything else that happens to us in our lives is layered with some degree of uncertainty. That’s why we pay attention to the weather report, buy insurance, and wish our friends “good luck.” Quantifying or giving a number value to this uncertainty is called probability. Using probability, chance occurrences can be analyzed mathematically and logically. Probability plays a central role in statistics when calculating hypothesis testing. We will use probability to decide if we should reject data based on very low probability. We begin by looking at the basics of probability and associated notations.

In probability, we deal with procedures that produce outcomes . An event is any collection of results or outcomes of a procedure. All possible outcomes of an event create a sample space (Triola & Iossi, 2018). An example of a procedure is to roll one six-sided die. The outcomes of rolling a six-sided die are {1, 2, 3, 4, 5, 6}. This set of numbers would be the sample space of the procedure. An event that could occur would be rolling an even die. This event’s sample space would be composed of the following outcomes: {2, 4, 6}. Calculating Basic Probability To find the probability of an event, the number of outcomes in an event is divided by the total outcomes in the procedure. When calculating probability, note the following properties. The notation P(event) is used to represent the probability of an event. All probabilities are a value between 0 and 1. A negative number and number larger than 1 are not possible for a measure of probability. The scale below breaks down the likelihood of probability values. Notice that the closer the value to 0, the less likely that event is to occur. The closer the probability to 1, the more likely the event is to occur (Triola & Iossi, 2018). Examples of Calculating Probability

Using the example of rolling a six-sided die given above, find the following probabilities: Example 1 - What is the probability of rolling a 3? Begin by determining the sample space of the event “rolling a 3.” It would consist of one outcome: {3}. The number of outcomes (1) would become the numerator. The denominator would be the total number of outcomes of the procedure (rolling a six-sided die), which would be 6. The probability of rolling a 3 would be 1/6 or 0.167. Example 2 - What is the probability of rolling an even number? Begin by determining the sample space of the event “rolling an even number.” It would consist of three outcomes: {2, 4, 6}. The number of outcomes (3) would become the numerator. The denominator would be the total number of outcomes of the procedure (rolling a six-sided die), which would be 6. The probability of rolling an even number would be 1/2 or 0.5. In some probability examples, you will be given the number of outcomes of a procedure and asked to calculate the probability. To calculate the probability, take the number of outcomes for an event and divide by the total outcomes possible. Example 3 - In a statistics course, 23 students passed the quiz with a 60% or higher while 5 students failed the quiz with a score lower than 60%. What is the probability that a student passed the quiz? In this example, 23 students passed the quiz, out of a total of (23 + 5) = 28 students. This produces the following probability.

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