A die marked 1,,2,,3 in red and 4,,5,,6 in green is tossed. Let A be event, ' number is even, &#

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14.Probability

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

Option A

Option B

Option C

Option D

Solution

When a die is thrown, the sample space ( $S$ ) is

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

Let $A:$ the number is even $=\{2,4,6\}$

$\Rightarrow P(A)=\frac{3}{6}=\frac{1}{2}$

$B:$ the number is red $=\{1,2,3\}$

$\Rightarrow P(B)=\frac{3}{6}=\frac{1}{2}$

$\therefore $ $A \cap B=\{2\}$

$P(A B)=P(A \cap B)=\frac{1}{6}$

$P(A) P(B)=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \neq \frac{1}{6}$

$\Rightarrow $ $P(A) \cdot P(B) \neq P(A B)$

Therefore, $A$ bad $B$ are not independent.

Standard 11

Mathematics

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