11Hindependent


 * Independent Events **


 * Events are ** independent ** if the outcome of one has no influence on the outcome of the other.
 * eg Two dice are rolled. The result on each dice is independent of the other


 * Events are ** dependent ** if the outcome of one does influence the outcome of the other.
 * eg A car driving too fast is more likely to be in an accident than a car driving more sedately. So the event that there is an accident is dependent on the speed of the car.
 * Note that other factors can be involved as well in this example. Two events being dependent doesn't exclude other events also being dependent


 * If two events, A and B, are ** independent **, then the probability of both A __**and**__ B happening is:
 * Pr(A Ç B) = Pr(A) × Pr(B)


 * ** Example 1 **
 * A dice is rolled twice. What is the probability that a 1 occurs on both dice rolls.

math . \qquad \qquad \qquad Pr(dice =1) = \dfrac{1}{6} math
 * ** Solution: **

math . \qquad \qquad \qquad Pr( \text{both dice = 1}) = Pr(\text{1st dice = 1}) \times Pr(\text{2nd dice = 1}) \\. \\ . \qquad \qquad \qquad Pr( \text{both dice = 1}) = \dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{36} math
 * the two dice rolls are independent so:


 * ** Example 2 **
 * Pr(A) = 0.2
 * Pr(B) = 0.5
 * Pr(A Ç B) = 0.8
 * Are events A and B independent?


 * **Solution:**
 * If A and B are independent, then
 * Pr(A Ç B) = Pr(A) × Pr(B)


 * Pr(A) × Pr(B) = 0.2 × 0.5 = 0.10
 * But Pr(A Ç B) = 0.8

.
 * so A and B are not independent (ie they are dependent)