Question

that at least two balls wl have different colors? 4. Prove that P(A1 UA2 U U An) Ση 1 P(Ai). This is known as Booles inequality and is sometimes called union bound. You may use without proof any formula proved in class. Hint: Use mathematical induction

Answer 1

To prove:

$i=1$

Proof:

Step 1:

For n = 1:

Thus, the result is true for n =1

Step 2:

Suppose the result is true for n. To prove it is true for n + 1

Step 3:

For the case n, we have:

(1)

Step 4:

By Addition Theorem:

(2)

P(AUB)- P(A) +PUB) - PAB)

Step 5:
Since union operation is associative, we get using (2):

(3)

Step 6:

By First Axiom of Probability:

(4)

Step 7:

Using (4), equation(3) becomes:

(5)

Step 8:

Simplifying RHS of (5), we get:

Thus, by mathematical induction, we get the result:

$i=1$