1) intergers a,b,c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that ad-bc is even?

Question

anonymous · Answer

the answer
 is 5/8

anonymous · Answer

ad- bc = EVEN means we must have

ad, bc both ODD
or
ad, bc both EVEN

in the first case we have 

a,b,c,d all odd

in the second case we just need one in each pair to be EVEN

anonymous · Answer

there are 1004 odd numbers from 0 to 2007 inclusive

there are 1004 even

anonymous · Answer

the probability of picking 4 odd numbers 

$$(1/2)^4$$

the probability of picking at least 1 even in each pair

$$= (1- (1/2))^2$$
$$=(3/4)^2$$

so we add them and get

$$\frac{1}{16} + \frac{9}{16} = \frac{10}{16} = \frac{5}{8}$$

anonymous · Answer

ok?