probability

A probability is a numerical measure of the chance of a particular event occurring.

It can be determined experimentally or by using logic.

overview

Probability contains:

• Discrete probability includes only a limited amount of outcomes, including the binomial distribution.
• Continuous probability includes a continuous range of outcomes, including the normal distribution.
• Sampling questions can include elements of both discrete and continuous probability methods, you need to distinguish what tools to use.

A trial can be repeated under the same conditions. A trial must result in one of the possible outcomes. All possible outcomes must be known before the trial begins.

An event is depicted with a capital letter. \for an event $A$, the probability of $A$ is shown as $\text{Pr}(A)$ and $0\le \text{Pr}(A)\le 1$.

The union of events $A$ and $B$ is shown as $A\cup B$ and means the situation where either event happens or both events happen. This is read as A or B.

The intersection of events $A$ and $B$ is shown as $A\cap B$ and means the situation where both events happen. this is read as A and B.

addition and multiplication rules of probability

The addition rule says $\text{Pr}(A\cup B)=\text{Pr}(A)+\text{Pr}(B)-\text{Pr}(A\cap B)$. This comes from adding both probabilities, and subtracting the overlap between them.

The multiplication rule says $\text{Pr}(A\cap B)=\text{Pr}(A)\cdot \text{Pr}(B|A)$. This is derived from a tree diagram. As you can see, the probability of both is equal to $\text{Pr}(A)$ and $\text{Pr}(B|A)$ multiplied together, where the second one is “probability of B given A”.

Two events $A$ and $B$ are mutually exclusive if $\text{Pr}(A\cap B)=0$. There is no possible overlap between the events.

Saying two events $A$ and $B$ are independent is equivalent to saying $\text{Pr}(A\cap B)=\text{Pr}(A)\cdot \text{Pr}(B)$ and $\text{Pr}(A|B)=\text{Pr}(A)$ and $\text{Pr}(B|A)=\text{Pr}(B)$. This means that the events are not related to each other.

A two-way table, aka Karnaugh map, is very useful to visualize the relationships between two events.

	A	A’
B	$\text{Pr}(A\cap B)$	$\text{Pr}(A^{\prime }\cap B)$	$\text{Pr}(B)$
B’	$\text{Pr}(A\cap B^{\prime })$	$\text{Pr}(A^{\prime }\cap B^{\prime })$	$\text{Pr}(B^{\prime })$
	$\text{Pr}(A)$	$\text{Pr}(A^{\prime })$	1

Note that $\text{Pr}(A\cup B)=1-\text{Pr}(A^{\prime }\cap B^{\prime })$.

conditional probability

the probability of an event occurring given that another event has already occurred is called conditional probability.

the probability of $A$ given that $B$ happened is depicted as $\text{Pr}(A|B)$, the probability of $A$ given $B$

to calculate conditional probability we use an equation derived from the multiplication rule of probability, $\text{Pr}(A|B)=\frac{\text{Pr}(A\cap B)}{\text{Pr}(B)}$

exercise - probability