Need help with the following Probability questions:

Problem1: There are two urns. The first urn contains 5 red and 7 green balls. The second urn contains 2 red and 2 green balls. We first draw a random ball from the first urn and put it into the second urn. Then we draw a random ball from the second urn.

1. What is the probability that the ball we drew from the second urn was red?
2. What is the probability that given that we drew a green ball from the second urn, the ball we put from the first urn to the second urn was green?

Problem2: Assume that we have 6 unfair coins. If weflip the i^th coin, then it shows heads with probability i/6.

1. We select one of the coins randomly andflip, and it shows heads. Find the conditional probability that the coin we selected was the third coin.
2. We flip the first, second and third coin once each. Let X be the random variable that denotes the total number of heads we get. Calculate E[X] and Var(X).

Problem3: We first draw 3 cards at random without replacement from a deck of 52 cards. We then draw another 3 cards at random without replacement from another deck of 52 cards. Find the probability that these six cards together do not contain any pairs.(Two cards are a pair if they have the same number, for example two sevens or two kings).

You don't need to compute the probability numerically in this problem, but the derivation of your final expression should be clear.

Hint 1: Denote the following event by A_i : the first selected i cards do not contain any

pairs. Then we want to find P(A₆). Note that A₆A₅A₄A₃A₂A₁.

Hint 2: You can also nd the probability directly using a counting technique.