Introduction to Probability

Types of Probability

The probability of an event E is the chance or likelihood that the event E will happen is denoted by $P\left(E\right)$.

Theoretical Probability

It is calculated by dividing the number of desired outcomes by the total number of all possible outcomes that could occur in theory.

The theoretical probaiblity that event E occurs is

Example

• You flip an ordinary coin twice.
• The sample space (a set showing all possible outcomes) is $\left\{HH,\ HT,\ TH,\ TT\right\}$.
• The probability that you get tails twice is $\frac{1}{4}$

Relative Probability

It is calculated by dividing the number of desired outcomes by the number of trials in an actual experiment.

The relative probability that event E occurs is

Example

• We toss an ordinary coin 200 times (# trials) and get heads 60 times (# observations).
• Based on this experiment, we conclude that the probability of heads is $\frac{60}{200}=0.30$
• Since the probabilities of 'Heads' and 'Tails', in this experiment, are not equally likely (i.e. the events have equal probability), then we consider this an unfair or biased coin.

In the long run with infinitely many trials, the relative probability will approach closer and closer to the theoretical probability (actual probability). This is called the Law of Large Numbers.

Subjective Probability

It is the probability that’s guessed by an individual based on their knowledge.

Example

• You believe that you will ace your exam after this session with a 70% probability.

Probability Language & Notation

Union

The union of 2 events A and B includes all the outcomes in A and B. This is denoted by $A\cup B$.

Example

$A$ is the event of rolling a number lower than 4 on a die.

$B$ is the event of rolling an odd number on a die.'

$A\cup B=$ {Event A or Event B}

Intersection

The intersection of 2 events A and B includes all the outcomes that are overlapping in both A and B. This is denoted by $A\cap B$.

Example

$A$ is the event of rolling a number lower than 4 on a die.

$B$ is the event of rolling an odd number on a die.

$A\cap B=$ {Event A and Event B}

Complement

The complement of Event A includes all the outcomes that are not in Event A. This is denoted by $A'$ or $A^c$ or $\overline{A}$.

Examples

$A$ is the event of rolling an even number on a die. $A'$ is the event of not rolling an even number on a die

$A$ is the event that you pass your exam

$A'$ is the event that that you do not pass your exam

$A$ the event that two people will get along

$A'$ is the event that two people will not get along

Example: Probability Language

Consider these sets:

$A = \{1, 2, 3, 4, 5\} \\ B = \{1, 2, 6\} \\ C = \{3, 5\}$

Find the following:

$A∩ B =$

$B ∩ C =$

$A ∪ B =$

$\overline{A}=$

Independent vs. Mutually Exclusive Events

Independent Events

Events A and B are independent if the occurrence of one does not affect the probability of the other to occur.

Example

Flipping "heads" from a coin does not affect the probability of rolling a "5" on a die; it's still P("5") = $\frac{1}{6}$

Mutually Exclusive Events (a.k.a. Disjoint)

Events A and B are mutually exclusive if they cannot both occur at the same time.

Note: $A\cap B=\varnothing$ ( the empty set)

Example 1

A $\to$ John was a home with his wife at 5pm

B $\to$ John was at the bank at 5pm

Example 2

A $\to$ Susan can't find her keys.

B $\to$ Susan spills her coffee.

Example: Independent Events

There are 4 black and 3 white balls in an urn. Two balls are drawn, one at a time, without replacement.

• Event A $\rightarrow$ First ball is black
• Event B $\rightarrow$ Second ball is black

(a) Are the events A and B independent? Mutually Exclusive?

(b) How do your answers change if we replace the balls each time?