Answer each question.

1. ProvethatP(EF^c)=P(E)P(EF)).
2. Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and7redballs. Ateachstage, aballisdrawn,anditscolor isnoted.Theballisthenreturned to the urn, along with an additional ball of the same color.Find the probability that the sample will contain exactly.
   1. 0whiteballs;
   2. 1whiteball;
   3. 3whiteballs;
   4. 2whiteballs.
3. Imagine you are a financial analyst at an investment bank.According to your research of publicly- traded companies, 60% of the companies that increased their share price by more than 5% in the last three years replaced their CEOs during the period. Atthesametime,only35%ofthecompaniesthatdidnotincreasetheir sharepricebymorethan 5% in the same period replaced their CEOs.Knowing that the probability that the stock prices growby more than 5% is 4%, find the probability that the shares of a company that fires its CEO will increase by more than 5
4. LetXbe the winnings of a gambler.Let p(i)=P(X= i)and suppose that p(0)=1/3;p(1)=p(1)=13/55;p(2)=p(2)=1/11;p(3)=p(3)=1/165.Compute the conditional probability thatthegamblerwinsi,i=1,2,3,giventhathewinsapositive amount.
5. A ball is drawn from an urn containing 3 white and 3 black balls.After the ball is drawn, it is replaced andanotherballisdrawn.Thisprocessgoesonindefinitely.Whatistheprobabilitythatofthefirst 4 balls drawn, exactly 2 are white?