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An Introduction to Probability

An Introduction to Probability

Author: Dan Laub

In this lesson, students will learn the basics of probability, and how it can be applied.

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Video Transcription

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Hi, Dan Laub here. And in this lesson, we're going to discuss An Introduction to Probability. But before we do so, let's discuss the objectives for this lesson.

The first objective is to understand the difference between experimental and theoretical probability. And the second objective is to understand the basic probability associated with the outcomes of experiments. So let's get started.

There exists two types of probability, theoretical and experimental. Theoretical probability is focused on forecasting how probable it is for an event to happen based on perfect probabilities. While the goal of experimental probability is describe how probable is that an event will occur based on experiments that had been done.

This lesson will focus on experimental probability. Experimental probability is important, because while theoretical probability can sometime tell us mathematically what the probability of an outcome is, actually conducting trials of experiments provides us with a realistic sense of what these probabilities actually are. One uses experimental probability to determine the probability that an outcome of a specific event will occur, when they may not know the probabilities associated with such outcomes. While one uses theoretical probability when the likelihoods of all possible outcomes are already know.

The idea behind conducting experiments is to find out more about the possible probabilities that exist for specific outcomes. When we conduct experiments in order to establish what the distribution of outcomes is, we refer these experiments as probability experiments. We refer to events as a group of outcomes of an experiment in which the experimenter may be focused on.

Some examples of probability experiments are surveying people to find out whether they might prefer one brand of soda over another. Or whether or not a basketball player is more likely to make a second free throw after making the first one. Or perhaps considering if a test group has a certain side effects related to a new prescription drug.

In the case of the survey, the outcomes would be Brand A, Brand B, or No Preference. And an event would be the response that a person conducting the survey would be interested in. In the case that the experimenters conducting market research and is looking at the characteristics of people interested in Brand A, that would be why they would focus on the Brand A outcome as an event as opposed to the other outcomes.

In the experiment pertaining to the probability associated with the basketball player making free throws, the outcomes would be make the shot or miss the shot. An event would be related to the question the experimenter is interested in answering, which in this case may be make the shot depending on whether or not they made the first one. We consider the probability of an outcome as the proportion that the outcome occurs in the probability experiment when considering all of the possible outcomes.

Such a probability of an event, which we will call Event A, is denoted by the term that you see here. The probability of event A is equal to P, parentheses, A. If we were interested in learning more about the probability of a specific event taking place, we look for something called favorable outcomes. These outcomes are those in which all outcomes related to an event, which we again will call Event A, actually do occur.

For example, suppose that you were playing a board game that involves rolling two, 6-sided dice to determine how many spaces you need to move. If you need to move eight spaces, there are the following combinations of rolls that will yield that outcome. It is important to realize that such outcomes aren't necessarily good, just that they are the outcomes that one is interested in.

In the case of playing a board game, one would not be interested in the outcomes of rolling two dice that yield seven spaces if they happen to need eight spaces to achieve a specific goal. When every one of the outcomes of an experiment is equally likely to occur, the probability of a given Event A, is equal to the number of favorable outcomes divided by the total number of possible outcomes.

As a simple example, consider a coin toss. There are two possible outcomes, heads and tails. And both have an equal probability of occurring. In this case, the probability of a coin landing heads up is equal to 1 divided by the total number of outcomes or 2.

Possible outcomes that an experimenter considers are based on the observed outcomes that occur as the experiment is conducted. In such an experiment, favorable outcomes are the outcomes that are related to the particular outcome that one is most interested in discovering more about. In this course, we will only focus on the types of experiments in which the outcomes are equally likely to each other.

Let's consider the example of a random experiment in which the experimenter has someone take out of a hat 1 of 20 small pieces of paper that each have a number written on them from 1 to 20. Since the person choosing the piece of paper cannot see which number is written on it, and we are assuming that the numbers are being picked randomly, the person has an equal chance of picking any one of the 20 pieces.

Such an experiment is designed to see if one number piece of paper happens to be chosen more often than the others. There are 20 possible outcomes in this experiment, with each having a 1 in 20 probability of occurring. So if the event we choose to focus on in this instance, all even numbers, we would consider it to be the favorable outcome and to have a probability 10 in 20.

Suppose that in another example, a researcher is interested in the game rock, paper, scissors. If a group of people are playing the game and a random selection of them are chosen, we would expect that the possible outcomes of an individual's choices are rock, paper, or scissors. Theoretically, we might expect each outcome to occur with equal probabilities. That is, each occurs 1/3 of the time.

But perhaps the researcher is curious as to whether or not that is actually the case. So in the instance of this experiment, if one is interested in how often a person chooses paper, the probability of that occurring is equal to this one favorable outcome divided by all three outcomes. Or in this instance, 1/3, or 1 out of 3.

So let's look back at our objectives to make sure we covered what we said we would. The first objective was to understand the difference between an experimental and theoretical probability, which we covered. And the second objective was to understand the basic probability associated with the outcomes of experiments. And we went through a few examples to illustrate that point.

So again, my name is Dan Laub. And hopefully, you got some value from this lesson.

Formulas to Know
Probability of A

P left parenthesis A right parenthesis space equals space fraction numerator n u m b e r space o f space o u t c o m e s space i n space A over denominator t o t a l space n u m b e r space o f space o u t c o m e s end fraction