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

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- 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.

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