1.An urn contains five red balls and one white ball. A ball is drawn and then it and another ball of the same color are placed back in the urn. Finally, a second ball is drawn.

a.What is the probability that the second ball is white?

I am pretty sure how to solve this, but if anyone find any mistake or awkward bit please let me know,

R-red, W-white

if first one was R(5/6) then second will be two cases-----R(6/7)

- -----W(1/7)

if first one was W(1/6)then second will be two cases-----R(5/7)

-----W(2/7)

so P(W)=(5/6)(1/7)+(1/6)(2/7)=7/42

my answer is 7/42,,,,is it correct?

b.If the second ball is white, what is the probability that the first was red?

using P(A and B)/P(A)=P(B?A)

in this case P(R ?W)= P(R and W)/P(W)

so (5/6)(1/7)/(7/42)=5/7

my answer is 5/7 I am not sure whether it is right..

2.A team of 2 players enters a quiz show, in which they have to answer 'true-false' questions. Each player will, independently, give the correct answer with probability p. Before the show they are trying to decide between the following strategies.

a.take turns answering questions.

b.Both consider each question and then either give the common answer if they agree or, if they disagree, flip a fair coin to determine the answer.

which, if either, is the better strategy?

3.A year after completing a certain 'quit smoking' class, 48% of the women in the class and 37% of the men had not resumed smoking. These people all attended a success party to celebrate their achievement. If 62% of the original class were male, use apporpriate results in probability theory to determine;

a.the percentage of the original class who attended the party

b.the percentage of those attending the party who were women.

4.A machine consists of four components, labelled 1,2,3,4 arranged as shown.

A --> 1 -->B -->3 -->C

A --> 2 -->B -->4 -->C

Each component has probability p of failure, independently of the others. For the machine to function, A must be connected to C by a path of working components. What is the probability that the machine functions? what is the range of permissible values for p if we require the probability of machine failure to be smaller than 0.05?