What is the probability formula?

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Probability for Statistics and Data Science

Probability for improved business decisions: Introduction, Combinatorics, Bayesian Inference, Distributions

03:40:27 of on-demand video • Updated January 2020

  • Understand probability theory
  • Discover Combinatorics
  • Learn how to use and interpret Bayesian Notation
  • Different types of distributions variables can follow
English [Auto] Hey everyone. Life is filled with uncertain events and often we must consider the possible outcomes before deciding. We ask ourselves questions like What is the chance of success and what is the probability that we fail to determine whether the risk is worth taking. Many CEOs need to make huge decisions when investing in their research and development departments or contemplating buyouts or mergers. By using probability and statistical data they can predict how likely each outcome is and make the right call for their firm. Some of you might be wondering what is this probability we're talking about. Essentially probability is the chance of something happening. A more academic definition for this would be the likelihood of an event occurring. The word event has a specific meaning when talking about probabilities. Simply put an event is a specific outcome or a combination of several outcomes. These outcomes can be pretty much anything getting heads when flipping a coin rolling a four on a six sided die or running a mile in under six minutes take flipping a coin for example. There isn't only one single probability involved since there are two possible outcomes getting heads or getting tails. That means we have two possible events and we need to assign probabilities to each one when dealing with uncertain events. We are seldom satisfied by simply knowing whether an event is likely or unlikely. Ideally we want to be able to measure and compare probabilities in order to know which event is relatively more likely to do so. We express probabilities numerically even though we can express probabilities as percentages or fractions conventionally. We write them out using real numbers between 0 and 1. So instead of using 20 percent or 1 5th we prefer point 2 All right. Now let us briefly talk about interpreting these probability values having a probability of 1 expresses absolute certainty of the event occurring and a probability of zero expresses absolute certainty of the event not occurring. You probably figured this out but higher probability values indicate a higher likelihood OK. As you can imagine most events we are interested in would have a probability other than 0 and 1. So values like point two point five and point six six are what we generally expect to see even without knowing any of this. You can tell some events are more likely than others. For instance your chance of winning the lottery isn't as great as winning a coin toss. That's why you can think of probability as a field that is about quantifying exactly how likely each of those events are on their own and that's what this course is going to teach you. So how about we start right away let's get into it generally the probability of an event a occurring denoted P of A is equal to the number of preferred outcomes over the total number of possible outcomes by preferred winning outcomes that we want to see happen. A different term people use for such outcomes is favorable similarly. Sample space is a term used to depict all possible outcomes going forward. We shall use the respective terms interchangeably. We will go through several examples to ensure you understand the notion well say event a is flipping a coin and getting heads. In this case heads is our only preferred outcome assuming the coin doesn't just somehow stay in the air indefinitely. There are only two possible outcomes. Heads or tails. This means that our probability would be a half. So we write the following P of getting heads equals 1 1/2 which equals point five. All right. Now imagine we have a standard six sided die and we want to roll a four once again. We have a single preferred outcome but this time we have a greater number of total possible outcomes. Six. Therefore the probability of this event would look as follows. P of rolling four equals one sixth or approximately zero point one six seven great. Events can be simple or a bit more complex. For example what if we wanted to roll a number divisible by three. That means we need to get either a three or a six. So the number of preferred outcomes becomes two however the total number of possible outcomes stays the same since the die still has six sides. Therefore we conclude that the probability of rolling a number divisible by 3 equals 2 over 6 which is approximately point 3 3 so far so good. Note that the probability of two independent events occurring at the same time is equal to the product of all the probabilities of the individual events. For instance the likelihood of getting the ace of spades equals the probability of getting an ace times the probability of getting a spade in a later lecture. We are going to define what we mean by independent. But for now let's observe some more examples of probability what about the probability of winning the U.S. lottery even though it sounds like something that is completely different. It actually follows the same idea. You take the number of preferred outcomes and divide it by all outcomes. Now the number of preferred outcomes we have would be equal to the amount of different tickets we bought. The total number of possible outcomes on the other hand is just something we will learn how to calculate less than an hour from now. For the moment just assume that there exists upward of one hundred seventy five million outcomes for the U.S. lottery therefore each individual ticket only has a probability of winning equal to one over one hundred seventy five million or approximately zero point 0 0 0 0 0 0 0 0 5. How would your chances improve if you bought two tickets. How about 5. I don't know about you but I like my odds of flipping a coin a lot more now that you know what probabilities are. Some of you might be wondering how and when we can use them in the next video we are going to do that by introducing expected values. Thanks for watching.