Examples

Independence and complements

Prove that these are logically equivalent statements:

Make sure you demonstrate both directions of each equivalency.

A bin contains 4 red and 7 green marbles. Two marbles are drawn.

Let be the event that the first marble is red, and let be the event that the second marble is green.

(a) Show that and are independent if the marbles are drawn with replacement.

(b) Show that and are not independent if the marbles are drawn without replacement.

Solution

(a) With replacement.

(1) Identify knowns.

Know:

(2) Compute both sides of independence relation.

Relation is

Right side is

For , have ways to get , and total outcomes.

So left side is , which equals the right side.

(b) Without replacement.

(1) Identify knowns.

Know: and therefore

We seek: and

(2) Find using Division into Cases.

Division into cases:

Therefore:

Find these by counting and compute:

(3) Find using Multiplication rule.

Multiplication rule (implicitly used above already):

(4) Compare both sides.

Left side:

Whereas, right side:

But so and they are not independent.