I am trying to solve the following problem:
Two dices are thrown.
$F_{e}$ = the event that the first die is even
$S_{4}$ = the event that the second die is 4
$\Sigma_{o}$ = the sum of the two dice is odd
Which of the events [1-4] are independent?
- $F_{e}$ and $S_{4}$
- $F_{e}$ and $\Sigma_{o}$
- $S_{4}$ and $\Sigma_{o}$
- $S_{4}$, $\Sigma_{o}$ and $F_{e}$ are mutually independent
Soliciting help to clarify my understanding.
$P(F_{e}) = \frac{1}{2}, P(S_{4}) = \frac{1}{6} and P(\Sigma_{o}) = \frac{1}{2}$
$P(F_{e} \cap S_{4}) = \frac{3}{36} = \frac{1}{12} = P(F_{e}) * P(S_{4})$ Hence independent
$P(F_{e} \cap \Sigma_{o}) = \frac{9}{36} = \frac{1}{4} = P(F_{e}) * P(\Sigma_{o})$ Hence independent
$P(\Sigma_{o} \cap S_{4}) = \frac{3}{36} = \frac{1}{12} = P(\Sigma_{o}) * P(S_{4})$ hence independent
[4] they are not mutually independent as
$P(\Sigma_{o} \cap S_{4} \cap F_{e}) \ne P(\Sigma_{o}) * P(S_{4}) * P(F_{e})$
- Did I do it correctly?
- Is there a better way to approach this problem?