Probability

Lesson · Beginner

Probability

Probability measures how likely an event is to occur.

A probability is always between 0 and 1:

0\le P(A)\le1

where $P(A)$ denotes the probability that event $A$ occurs.

A probability of

0

means the event is impossible, while a probability of

1

means the event is certain.

The sample space is the set of all possible outcomes of an experiment.

It is usually denoted by

S.

An event is a collection of outcomes from the sample space.

For example, when rolling a standard die,

S=\{1,2,3,4,5,6\}.

If $A$ is the event that an even number is rolled, then

A=\{2,4,6\}.

If all outcomes are equally likely, then

P(A)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}

or equivalently,

P(A)=\frac{|A|}{|S|}

For the die example,

P(A)=\frac{3}{6}=\frac12.

The complement of $A$ , denoted by $A^c$ , is the event that $A$ does not occur.

Since either $A$ occurs or it does not,

P(A^c)=1-P(A)

Equivalently,

P(A)+P(A^c)=1

Events $A$ and $B$ are independent if

P(A\cap B)=P(A)P(B)

An integer is chosen uniformly at random from

\{1,2,\ldots,n\}.

Let

A=\{\text{the chosen integer is divisible by }2\}

and

B=\{\text{the chosen integer is divisible by }3\}.

Find all positive integers $n>1$ for which $A$ and $B$ are independent.