Dependent and Independent Events

Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:

$$ \text P ( \text A  |  \text B)  =  \text P (\text A) $$

or

$$ \text P (\text B  |  \text A)  =  \text P (\text B) $$

The events are said to be dependent if this condition is not satisfied.

Example 1: Independent Events

Suppose you rolled a die and flipped a coin. The probability of getting any number face on the die does not influence the probability of getting a head or a tail on the coin. The two events are independent.

Example 2: Dependent Events

Suppose you have a bag containing 8 green balls and 5 blue balls. If you draw a blue ball without replacing it, the probability of drawing another blue ball on your second attempt is greatly changed because you drew a blue ball the first time. Such events are said to be dependent.

If two events are independent:

$$ \text P (\text {AB})  = \text P (\text A)  × \text P (\text B) $$

Question

Mike Jamerson tosses a coin twice. What is the probability of getting a head on the first toss and a head on the second toss?

A. \(\frac{1}{2}\).

B. \(\frac{1}{4}\).

C. \(\frac{2}{3}\).

Solution

The correct answer is B.

The coin has no memory. As such, the first outcome has no influence on the second outcome. Therefore:

$$ \begin {align*}
\text P (\text {Head and Head})  &  =  \text P (\text H) × \text P (\text H) \\
&  = \cfrac  {1} {2} × \cfrac  {1} {2} \\
&  =  \cfrac  {1} {4}  \\
\end {align*} $$