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1. Overview of Probability
In this lesson, we’ll learn what a probability is, and learn about important probability concepts like events, complements and expected value. We’ll also learn how to calculate basic probabilities.
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Lesson Goal (00:34)
The goal of this lesson is to learn what probability is, and calculate some simple probabilities.
Events and Probabilities (00:40)
An event is an outcome or combination of outcomes that can occur from a process. For example, if we roll a dice, possible events include rolling a 4, rolling an even number, and so on. When we roll a dice, any valid event will either occur or not occur.
A probability is a numeric value that measures how likely it is that some event of interest will occur. To calculate the probability of an event, we divide the number of outcomes where the event occurs by the total number of possible outcomes. For example, when we roll a dice, there are 3 possible even numbers out of 6 possible outcomes, so the probability of rolling an even number is 3/6, or ½.
Probability Properties (03:44)
All probabilities have a value between 0 and 1. An event with a probability of zero will definitely not occur. An event with a probability of one will definitely occur. The higher the probability of an event, the more likely it is to occur. We can express probabilities using fractions, decimals, or percentages.
The complement of an event is the set of outcomes where the event does not occur. For example, the complement of rolling a two is rolling any number other than two. An event and its complement cannot occur at the same time, but one must always occur. As a result, the probability of an event and its complement add to one, and we can find the probability of a complement by subtracting the event’s probability from one.
Expected Value (05:51)
The expected value of a variable is the predicted value of that variable, based on the probability and value of each possible outcome. To calculate expected value, we multiply the value of each outcome by its probability, and add the result for each possible outcome.
As we adjust the value of each possible event, or the probability of each possible event, the expected value changes. We can use the expected value to determine if a process, such as a dice game, has any value to us.
In this course, we're going to learn about the fundamental principles of probability.
Probability is an area of statistics with many applications.
In this course, we'll learn how to calculate probabilities, learn about different types of probabilities, and use probability to solve various problems.
In this first lesson, we'll learn what probability is and calculate some simple probabilities.
Before we introduce the concept of probability, we need to understand the concept of events.
Let's consider a single roll of a fair six-sided die.
There are six possible outcomes.
Roll a one, a two and so on up to six.
An event is simply an outcome or a combination of outcomes that can occur from a process such as this.
Note that events must refer to specific outcomes.
In our case, rolling that die is not considered an event. Instead, the event refers to possible outcomes from rolling the die.
To understand this better, we'll define three possible events that can occur when we roll a die. Event A occurs when we roll a two.
Event B occurs when we roll any even number.
Event C occurs when we roll a value of four or less.
When we roll a die, none, some, or all of these events might occur.
Note that each individual event will either occur or not occur.
In other words, it's not possible for an event to partially occur.
Now let's consider a probability.
A probability is simply a numeric value that measures how likely it is that some event of interest will occur.
We can calculate the probability of an event by dividing the number of outcomes of interest by the total number of possible outcomes.
Let's do this now for our three events.
We'll denote probabilities using the letter P.
Event A occurs if we roll a two.
Therefore, there's only one outcome that makes event A occur.
There are six possible outcomes. So the probability of A or P of A is one out of six.
Event B occurs when we roll an even number.
There are three outcomes, where this event occurs. Two, four, and six.
Again, there are six possible outcomes. So the probability of B is 3/6, which we can reduce to 1/2.
Finally, Event C occurs when we roll a four or less. So there are four outcomes where this event occurs.
As before, there are six possible outcomes.
So the probability of C is 4/6, which we can reduce to 2/3.
Now that we understand what a probability is, let's consider some of the properties of probabilities.
The probabilities we just calculated all had fractional values below one. In fact, all probabilities have a value between zero and one.
If an event has a probability of zero, it means that the event will definitely not occur.
If an event has a probability of one, it definitely will occur.
The higher the probability of an event, the more likely it is that the event will occur.
Note that we can express probabilities using fractions, decimals, or percentages.
Next, let's learn about the complement of an event.
The complement of an event is the set of outcomes, where the event does not occur.
Let's consider an example to make this more clear.
Previously we said that Event A occurs when we roll a two on our die. This means that the complement of A occurs when we roll any number other than two.
We rate this as A dash.
Anytime we roll a die, either Event A or Event A dash will occur.
It's not possible for both to occur at the same time, but it's also not possible for neither event to occur.
As a result, we can say that the probability of A plus the probability of A dash must be one.
We can find the probability of A dash by subtracting the probability of A from one.
In our case, this is one minus 1/6 which is 5/6.
This makes sense as there are five outcomes where we don't roll a two out of the six possible outcomes.
Finally, we'll learn about the concept of expected value.
The expected value of a variable is based on the probability and value of each possible outcome. Again, we'll consider Event A.
We'll assume that rolling a die forms a part of a game.
If we roll a two, we win $5.
If we roll any other number, we lose $1.
The expected value will tell us how much money we can expect to win or lose if we play this game repeatedly for a long time.
To calculate the expected value, we use this formula.
It might look complicated, but it simply means that for each possible outcome of the game, we multiply the value of the outcome by its probability.
We then add the result together for each possible outcome, as indicated by the Greek letter Sigma, which represents a sum.
Our game has two possible outcomes, win or lose.
We know that the probability of winning is 1/6 and the value of winning is five.
Therefore the expected value of winning is five times 1/6 which is 5/6.
The probability of losing is 5/6. And the value of losing is negative one as we lose a dollar in this situation.
The expected value of losing is therefore negative one times 5/6 which is negative 5/6.
If we add these values together, we get zero, which is the expected value of playing this game.
The expected value of the game tells us that we won't win or lose money overall if we play this game repeatedly for a long time.
If we play this game only a few times, we might get lucky or unlucky and win or lose some money, but we shouldn't expect this to continue permanently.
Let's see how the expected value formula changes when we change some of the parameters of the game.
For example, if we win $10 every time we roll a two, this increases the expected value when we win.
We can see that the expected value is now 5/6 which tells us that we will win 5/6 of a dollar on average, every time we play this game.
Let's change the win amount back to $5 and change the lose amount to $2.
Now the expected value of the game is negative 5/6.
This means we can expect to lose 5/6 of a dollar every time we play this game.
As we can see the expected value of a problem, depends on both the probability of each possible outcome and the values associated with those outcomes.
Now that we've introduced the most important probability concepts, we'll stop the lesson here.
In the next lesson, we'll learn how to use a sample space.