Probability

probability-theory

Definition

Probability

Probability is a measure of uncertainty of the occurrence of an event, with an event being all desired outcomes out of all possible outcomes.

The probability $P(A)$ of an event $A$ out of all outcomes $\Omega$, where each event is equally likely, is given by:

$$P(A)=\frac{\#\text{of desired outcomes}}{\#\text{of all outcomes}}=\frac{|A|}{|\Omega |}$$

Examples

Fair Coin Toss

What is the probability of getting exactly 1 head out of 3 tosses?

Sample Space: Listing all $8$ possible outcomes:

$$\Omega =\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$$

Event: There are 3 outcomes in the “exactly 1 head”event

$$A=\{HTT,THT,TTH\}\subseteq \Omega$$

Probability: With all outcomes being equally likely, we get:

$$P(A)=\frac{|A|}{|\Omega |}=\frac{3\text{outcomes with}1\text{head}}{8\text{possible outcomes}}=\frac{3}{8}\approx 37.5\%$$