A die is thrown. If E is the event 'the numbe | vedclass

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Question

(A) The sample space of throwing a die is $S = \{1, 2, 3, 4, 5, 6\}$.The event $E$ (number is a multiple of $3$) is $E = \{3, 6\}$,so $P(E) = \frac{2}

Vedclass · Accepted Answer

Yes,they are independent.

Vedclass · Answer

(A) The sample space of throwing a die is $S = \{1, 2, 3, 4, 5, 6\}$.The event $E$ (number is a multiple of $3$) is $E = \{3, 6\}$,so $P(E) = \frac{2}{6} = \frac{1}{3}$.The event $F$ (number is even) is $F = \{2, 4, 6\}$,so $P(F) = \frac{3}{6} = \frac{1}{2}$.The intersection $E \cap F$ (number is a multiple of $3$ and even) is $E \cap F = \{6\}$,so $P(E \cap F) = \frac{1}{6}$.Two events are independent if $P(E \cap F) = P(E) 	imes P(F)$.Here,$P(E) 	imes P(F) = \frac{1}{3} 	imes \frac{1}{2} = \frac{1}{6}$.Since $P(E \cap F) = P(E) 	imes P(F)$,the events $E$ and $F$ are independent.

$A$ die is thrown. If $E$ is the event 'the number appearing is a multiple of $3$' and $F$ is the event 'the number appearing is even',then find whether $E$ and $F$ are independent?

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