Show that the results are the same despite the distribution

[rad6210]

Suppose a box contains tickets, each labeled by an integer. Let X,Y, and Z be the results of draws at random with replacement from the box: Show that, no matter what the distribution of numbers in the box, P(X+Y is even) >= 1/2

I have no idea how I would go about "showing" something like this so any help would be great thank you!

 

[arthur ohlsten]

you obtain a even number if both choices were even or both odd
let x = probability of even choice and y probability of odd choice

then probability of even choice is xx+yy this is P[x+y] being even
but y=1-x or
P[x+y]= x^2+[1-x]^2
P[x+y]=x^2+1-2x+x^2
P[x+y]=2x^2-2x+1

to find P[x+y] minimum take derivative and set to 0
dP[x+y] / dx = 4x-2
0=4x-2
x=1/2 or

P[x+y] is a minimum at x or P(e)=1/2 and is
P[x+y]=2[1/4]-2[1/2] +1
P[x+y]=1/2 then
P[x+y]>= 1/2 answer

Arthur