In this lesson, our instructor Vincent Selhorst-Jones teaches all about Probability. Youll learn the definition of sample space and learn how to predict the probability of an event. Vincent will explain why equally likely is also important to consider. Mutually exclusive events, overlap, complement of E, independent events, and conditional events are explained in detail. Then youll have the chance to work through five practice examples.
In the lesson on Counting, we defined the term event to simply mean something happening. For example, if we flipped a coin, we might name an event E that is the event of the coin coming up heads. Of course, there might be possibilities
other than the event occurring. We call the set of all possible outcomes the sample space. If we want, we can denote the sample space with a symbol, such as S. In the coin example, there are two possible outcomes in the sample space S: heads and
Let E be an event and S be the corresponding sample space. Let n(E) denote the number of ways E can occur and n(S) denote the total possible number of outcomes. Then if all the possible outcomes in S are equally likely, the probability of
event E occurring (denoted P(E)) is
Equivalently, using words,
Notice that in the previous definition, it was assumed that all the possible outcomes were "equally likely". If this isn't true, the above does not work. Happily, we are almost certainly not going to see any problems that don't involve equal likelihood
of all the outcomes. It might be described in the problem as `fair', `random', or something else, but we can almost always assume that all possible outcomes are equally likely at this level in math.
We can represent probability as a fraction ([1/2]), a decimal (0.5), or a percentage (50%). In any case, a probability is always between 0 and 1, inclusive. The larger the value, the more likely. We can also interpret probability as the ratio of the
event happening over a large number of attempts. For example, if we flip a coin a million times, we can expect about half of the flips to come out heads. (P(Eheads) = [1/2])
Given two mutually exclusive events A and B, the probability of either one (or both) occurring (A∪B) is given by
P(A ∪B) = P(A) + P(B) .
If the events are not mutually exclusive, we have to take the overlap into account. We can represent where A and B overlap with A ∩B. Then
P(A∪B) = P(A) + P(B) − P(A∩B).
If we have some event E, we can talk about the event of E not occurring. We call this the complement of E, denoting it as Ec. [Other textbooks/teachers might denote it ―E or E′.] The probability
of an event's complement occurring is
P(Ec) = 1 − P(E).
Two events are independent if they are separate events and the outcome of either one does not affect the other. Given two independent events A and B, the probability of both events occurring is
P(A and B) = P(A) ·P(B).
If the events are not independent (the outcome of one does affect the other) and we want to find the chance of them both occurring, we need the idea of conditional probability. We denote the conditional probability of B occurring if A does
occur as P(B | A). (We can interpret this as the probability of B happening if we are guaranteed that A will happen.) Then, given two events A and B, where the outcome of A affects the outcome of B, the probability of both events occurring is given by
P(A and B) = P(A) ·P(B | A).
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.