4-4 vs 0-4

Those two dominoes were the best options. There were no ideal choices but when placed in a situation like this, you have to do the math to figure out your best odds. Play the odds. When all the choices are bad, you must take the time to figure out which bad choice is your best choice. If you suggested anything other than the 4-4 or 0-4, this conversation maybe a little over your head. That being said, let's look at the two choices.

You already know this is a sticky proposition. If your opponent does not have the 4-2, you live to make another play. Of course, you are hoping for the 6-4 but the 4-1 also allows you to make another play. The danger of downing the 4-4 is you risk passing your opponent. How often have you had four of a kind, downed the double and your opponent drew five or six bones from the yard and built a good hand? You would only down the 4-4 hoping to win the game on your second play with the 6-2. It would be practically impossible to miss your opponent if he gets a chance to make a second play especially if he draws several bones from the yard. If he draws or plays the 4-1, you will find yourself in a situation where you must play the 6-1 which will give him only one opportunity to go out and that is with the 6-6 because getting behind the house would certainly create way too many opportunities for him to win the game. Although he would have an opportunity to win the game, if he does not draw the 6-4, you have will not have an opportunity to win the game on your second play. That is a major consideration whether you should down the 4-4.

Downing the 0-4 gives your opponent two chances to win the game, the 1-0 and the 6-0. If you are fortunate enough to miss him, you will automatically win the game on your 2nd play. So that versus giving him one opportunity to win the game by downing the 4-4 and giving yourself one opportunity to win the game if you missing him.
You may say "first things first," give him one opportunity and not two. In my opinion, that is the "dog barking at the car" mentality. It will be practically impossible for you to win the game if you miss him downing the 4-4 because you risk passing him which is never good in a situation like this.

4-4 - gives him one opportunity to win but if you miss, you give yourself only one opportunity (risk passing him on the first play which could be bad).
0-4 - gives him two opportunities to win but if you miss, you will automatically win.

Bottom Line - In all sincerity, the 0-4 is the best choice.

NOTE: I lost the game by downing the - 0-4 because he had the 6-0. Had I downed the 4-4, my opponent would have passed (he gave me a ss of his hand afterward).