Winning the Lottery

In a far-away land, the lottery consists of four balls numbered $1$ to $4$, which are placed in a bag.
To enter, you choose one number.
To win, your number must match the number that is drawn from the bag.

What is the chance of winning this lottery?
 
The people running the lottery in this far-away land decide that it is too easy to win.  So, they change their lottery game.
In the new lottery, there are still four balls numbered $1$ to $4$, which are placed in a bag.
Now, to enter, you choose two numbers.
To win, your numbers must match (in any order) the two numbers that are drawn from the bag.
 
What is the chance of winning this new lottery?
Have the organisers made it harder to win compared with their original version?
 
Can you create your own version of the lottery which would also be harder to win than the first game? 
How do you know that your game is harder?
 
You may like to use our lottery simulator to try out your ideas: