# A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event of the number being even, and B be the event of the number being red. Are A and B independent?

$\begin{array}{1 1} \text{A and B are independent events} \\ \text{A and B are not independent events} \\ \end{array}$

Toolbox:
• If A and B are independant events, $$P(A\cap\;B)=P(A)\;P(B)$$
The sample space for a roll of die can be expressed as follows: S = $\begin{Bmatrix} 1&2&3&4&5&6 \end{Bmatrix}$
Let A be the event where an even number appears.
A = $\begin{Bmatrix} 2&4&6 \\ \end{Bmatrix}$
Then P(A) = $\large \frac{3}{6} = \frac{1}{2}$
Let B be the event that a number in red appears on the die.
B = $\begin{Bmatrix}1&2&3 \\ \end{Bmatrix}$
Then P(B) = $\large\frac{3}{6} = \frac{1}{2}$
We can see that A $\cap$ B = $\begin{Bmatrix} 2\\ \end{Bmatrix}$
Therefore, P (A $\cap$ B) = $\large\frac{1}{6}$
We know that if A and B are independant events, $$P(A\cap\;B)=P(A)\;P(B)$$
$\Rightarrow P(A) \; P(B) = \large \frac {1}{2}$$\times \large \frac{1}{2} = \frac{1}{4} \neq$ P(A $\cap$ B).
Therefore A and B are NOT independent events.
edited Apr 21, 2016 by pady_1