Independent event

Two events are independent if whatever happesn does not affect the other one at all.

The probability of happening of two independent events is the product of the probability of both happenings. $p (X = x & Y = y) = p (X = x) \cdot p (Y = y)$

Example

The probability to get a $6$ in each of two independent dices $A$ and $B$ is: $p (X = 6 & Y = 6) = p (X = 6) \cdot p (Y = 6) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$

Example

The probability of rain one day in February in Sevilla is $35 %$ , and that the Betis wins is of $75 %$ .

What is the probability for not rain in Sevilla and for the Betis wins?

$P (not rain) = 1 - P (rain) = 0.65 P (not rain) \cdot P (Betis wins) = 0.65 \cdot 0.75 = 0.49 \to 49 %$