This is another set of questions that I've seen often and think are worthwhile covering. Any help is appreciated. Thank you.

I

If A and B are events in a probability space such that 0 < P(A) = P(B) = P(A intersect B) < 1, which of the following cannot be true?

ans a) A and B are independent

b) A is a proper subset of B

c) A != B

d) A intersect B = A union B

e) P(A)P(B) < P(A intersect B)

I thought the answer would be C since P(A) = P(B) = P(A intersect B)...

II This type of question is very very common

If x, y and z are selected independently and at random from the interval [0,1], then probability that x >= yz is

ans a) 3/4

b) 2/3

c) 1/2

d) 1/3

e) 1/4

Intuitively I know that yz is most likely smaller than x but I cannot go beyond saying that the answer is over 1/2.

III Another very common type

If x is a real number and P is a polynomial, then lim h->0 [P(x + 3h) + P(x - 3h) - 2P(x)]/h^2 =

a) 0

b) 6P'(x)

c) 3P''(x)

ans d) 9P''(x)

e) infinity

I saw another problem where they asked the same except the limit was lim h->0 [P(x+h) - P(x-h)]/h and I intuitively answered correctly, that the answer was 2P'(x). What is the procedure for this type of question?

IV

Consider the sytem of equations

ax^2 + by^3 = c

dx^2 + ey^3 = f

Where a, b, c, d, e and f are real constants and ae != bd. The maximum possible number of real solutions (x,y) of the system is

a) none

b) one

ans c) two

d) three

e) five

I thought a system of equations would have only 0, 1 or infinitely many solutions...