Urn A contains R red balls and W white balls, urn B contains r red balls and w white balls.

a.If an urn is selected at random and a ball drawn, what is the probability that the ball is red?

b.If a ball is drawn from urn A and put into urn B, and then a ball is drawn from urn B, what is the probability that the second ball is red?

c.If A and B are independent events, then prove that A^C (A raise to C) and B^C (B raise to C) are also independent.

a. R/(R+W) + r/(r+w) = ?

It's a start.

a. To find the probability that a ball drawn at random is red, we need to consider the total number of red balls and the total number of balls in both urns.

The total number of balls in both urns is the sum of the balls in each urn: (R + W) + (r + w) = (R + r) + (W + w).

The total number of red balls is R + r.

Therefore, the probability of drawing a red ball is (R + r) / ((R + r) + (W + w)).

b. If a ball is drawn from urn A and put into urn B, the number of red and white balls in urn B will change. After adding a ball from urn A to urn B, the number of red balls in urn B becomes r+1, and the total number of balls becomes (R+r+1)+(W+w).

To find the probability that the second ball drawn from urn B is red, we need to consider the new number of red balls and the new total number of balls in urn B.

The number of red balls in urn B after adding a ball from urn A is r+1.

The total number of balls in urn B after the addition is (R+r+1)+(W+w).

Therefore, the probability of drawing a red ball from urn B after the addition is (r+1) / ((R+r+1)+(W+w)).

c. To prove that events A and B are independent, we need to show that the probability of the intersection of the two events is equal to the product of their individual probabilities.

Let's assume that events A and B are independent.

The independence of A and B means that P(A and B) = P(A) * P(B).

To evaluate P(A and B), we consider the probability of the intersection of the two events.

P(A and B) = P(A) * P(B)

Using the complement rule, we can represent A^C (complement of A) and B^C (complement of B).

P(A^C and B^C) = P(A^C) * P(B^C)

Using the complement rule, we can write P(A^C) = 1 - P(A) and P(B^C) = 1 - P(B).

Substituting these values in the expression, we get:

P(A^C and B^C) = (1 - P(A)) * (1 - P(B))

Simplifying, we have:

P(A^C and B^C) = 1 - P(A) - P(B) + P(A) * P(B)

Now, let's evaluate the expression (1 - P(A)) * (1 - P(B)):

(1 - P(A)) * (1 - P(B)) = 1 - P(A) - P(B) + P(A) * P(B)

We can see that the expression (1 - P(A)) * (1 - P(B)) is equal to P(A^C and B^C).

Therefore, we have demonstrated that if A and B are independent events, then A^C and B^C are also independent.