Probability is a wide and deep subject. I’m going to present just the basics, which is a very subjective statement. This post will serve as a reference for you to use for more “advanced” topics. And yes “advanced” is also subjective.
Simple Probability
A very basic way of describing the probability of an event is the number of ways the event can occur divided by the number of outcomes in the sample space.
\(\text{E: event}\)
\(\text{S: sample space}\)
\[P(E)=\frac{n(E)}{n(S)}\]
Let’s look at a quick example with a Venn diagram.
\(\text{n(E) = 13}\)
\(\text{n(S) = 69}\)
The box represents our sample space. The circle represents our event E. Can you tell where the 56 came from? Also, note that the relative area of sections of Venn diagrams are arbitrary and don’t necessarily represent the number of elements in them.
So the probability
\(P(E)= \frac{n(E)}{n(S)} = \frac{13}{69}\)
And/Or
In mathematics, the notion of “and”/”or” is slightly different from how we use them in English. Well, really the problem is with the “or”.
A person asks for a “ham or turkey” sandwich. We would understand that this person would be happy with a single ham sandwich or a single turkey sandwich and would gladly pay you for the sandwich. However, if we presented this person with both a ham and turkey sandwich, we might find it difficult to convince this person to pay for both sandwiches.
A mathematical “or” is completely fine with both happening.
Event A or Event B, is satisfied when A happens, B happens, or both A and B happens.
Event A and Event B, is only satisfied when A happens and B happens.
With that being said there is a strong connection to this and/or with Unions/Intersections. However, since English is kind of messed up, this will not always %100 hold. We just have to do our best to tell if we are talking about a strictly mathematical “Or” versus a strictly English “Or”.
Unions (or)
I’m going to first take a Venn diagram approach to Unions and then a more formal definition.
Consider a Venn diagram of two sets.
Then a visual representation of the union of A and B would be
The formal way
“The Union of A and B is the set of outcomes x such that x is an outcome of A or x is an outcome of B”
\(A \cup B = \{x|x \in A \text{ or } x \in B\}\)
It should be noted that just because an outcome of both A and B that we will not count it twice.
Intersection (and)
The Venn diagram of the intersection of A and B is
The more formal way
“The Intersection of A and B is the set of outcomes x such that x is an outcome of A and x is an outcome of B”
\(A \cap B = \{x|x \in A \text{ and } x \in B\}\)
Examples
Consider the following Venn diagram
\(n(S)=68\)
\(n(A)=21\)
\(n(A’)=47\)
\(n(B)=42\)
\(n(B’)=26\)
\(P(A)=\frac{21}{68}\)
\(P(B)=\frac{42}{68}\)
\(P(A\cup B)=\frac{57}{68}\)
\(P(A\cap B)=\frac{6}{68}\)
\(P(A’\cap B’)=\frac{11}{68}\)
\(P(A’\cup B’)=\frac{62}{68}\)
\(P(A\cap B’)=\frac{36}{68}\)
\(P(A’\cap B)=\frac{15}{68}\)
I will leave it up to you to determine how each was determined.
Important Formulas
I can’t stress how important these formulas are. As a mathematician, I take these formulas for granted. I know them and can ramble them off at any time of the day, and use them too.
If you don’t already know these, I would suggest writing them down on a small piece of paper and taking them with you. Continue to keep them with you and look at them until you KNOW them.
Probabilities are between 0 and 1 inclusively. Yes, some things never happen and some things always happen. Don’t get too complicated with it. The point, it is a red flag if, probabilities are negative or greater than one.
\(0 \leq P(E) \leq 1\)
The relationship between Intersection and Union, colors help.
\(P(A \color{red}{\cup} B)=P(A)+P(B) – P(A \color{blue}{\cap} B)\)
\(P(A \color{blue}{\cap} B)=P(A)+P(B) – P(A \color{red}{\cup} B)\)
Compliments can be soo useful its almost stupid, one formula but also kind of three
\(1=P(E)+P(E’)\)
\(P(E)=1-P(E’)\)
\(P(E’)=1-P(E)\)
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