Examples

Independence and complements

Prove that these are logically equivalent statements:

• and are independent
• and are independent
• and are independent

Make sure you demonstrate both directions of each equivalency.

Solution

(1) Assume and show :

Divide into the cases:

Apply the assumption:

Algebra:

Negation rule:

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(2) Assume and show :

Reason the same way as above, but apply the new assumption to break up the second term instead of the first.

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(3) Show that and are equivalent:

To do this, simply notice that in the first equivalence we can replace with and with . Use too.

Independence by hand: red and green marbles

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.

Identify knowns:

• Know:
• Know:

Now compute both sides of independence relation:

The right side is .

For , we have ways to get , and total outcomes. So left side is , which equals the right side.

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(b) Without replacement. This is a bit harder.

(1) Identify knowns:

Know: and therefore

We seek: and

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(2) Find using Total Probability (Division into Cases):

Find RHS factors by counting, then compute:

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(3) Find using multiplication rule:

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(4) Compare both sides:

Left side: .

Right side:

But so and they are not independent.