# A fair coin and an unbiased die are tossed. Let A be the event - head appears on the coin and B be the event 3 on the die. Check whether A and B are independent events or not.

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

Toolbox:
• If A and B are independant events, $$P(A\cap\;B)=P(A)\;P(B)$$
The sample space for a coin toss and a roll of die can be expressed as follows:
S = $\begin{Bmatrix} 1H & 2H & 3H & 4H&5H &6H \\ 1T & 2T & 3T& 4T& 5T & 6T \end{Bmatrix}$
Let A be the event where a head appears.
A = $\begin{Bmatrix} 1H & 2H & 3H & 4H&5H &6H \\ \end{Bmatrix}$
Then P(A) = $\large \frac{6}{12} = \frac{1}{2}$
Let B be the event that a 3 appears on the die.
B = $\begin{Bmatrix} 3H & 3T \\ \end{Bmatrix}$
Then P(B) = $\large\frac{3}{12} = \frac{1}{6}$
We can see that A $\cap$ B = $\begin{Bmatrix} 3H\\ \end{Bmatrix}$
Therefore, P (A $\cap$ B) = $\large\frac{1}{12}$
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}{6} = \frac{1}{12} =$ P(A $\cap$ B).
Therefore A and B are independent events.