Theory Of Probability

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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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Caution: Make sure to always calculate the total in the denominator of your probability. In the example above, we needed to add the number who passed and the number who failed to calculate the total number of students. Tables can also be used to organize data for several variables. To calculate the probability of an event, choose the cell which contains the information and divide it by the total event that you are observing. Examples 4 and 5 A group of 50 people were asked the following question, “Do you prefer summer or winter?” The results are broken down into female and male categories in the table below. Summer / Winter / Total Male - 18 / 8 / 26 Female - 17 / 7 / 24 Total - 35 / 15 / 50 Example 4 - What is the probability that a randomly chosen person prefers winter? In this problem, we are looking at all 50 individuals. Since 15 total people prefer winter, the probability that a randomly chosen person prefers winter would be Example 5 - What is the probability that a randomly chosen female prefers summer?

In this example, we are only looking at the female row of 24 individuals. Since 17 females prefer summer, the probability that a randomly chosen female prefers summer would be: The Complement of an Event In statistics, we sometimes need to find the probability that an event does not occur. The complement of an event A , denoted by , consists of all outcomes in which event A does not occur. To find the complement’s size, take the total size of the procedure and subtract the number of elements in A . Knowing and calculating the complement may make calculating the probability of the original event easier (Triola & Iossi, 2018). Example 6 - In a statistics course, 25 out of 28 students completed their homework. How many students did not complete their homework? In this example, the event, A , would be the students who completed their homework. Therefore, the complement would be the students who did not complete their homework. This means that 28 - 25 = 3 students did not complete their homework. The probabilities would be:

These examples illustrate the basic building blocks of probability theory and associated notations using simple events. As you dive further into statistics, you will learn how to combine events to build more complex outcomes. This includes combining events into compound events to find the probability that several events occur at the same time. Additional Readings - Dice Control in Craps One popular casino game is craps. This game requires the player to roll two dice. Depending on the outcome of the dice, they either win or lose money. Read this article, Golden arm: A probabilistic study of dice control in craps to gain a better understanding of this game and how probability can be applied to this game. References Smith, D. R., & Scott III, R. (2018). Golden arm: A probabilistic study of dice control in craps. UNLV Gaming Research & Review Journal, 22 (1), 29-36. https://search.ebscohost.com/login.aspx? direct=true&AuthType=ip,shib&db=iih&AN=133727080&site=eds- live&custid=s9076023 Triola, M. F., & Iossi, L. (2018). Elementary statistics (13 th ed.). Pearson.

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Theory Of Probability

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