The importance of the doubling dice cannot be overly stressed and the equity calculations are similar to online poker calculations in poker hands like you would see in the cash games at poker sites. Without it, backgammon can be a relative game of chance within a 30-40% margin.
With it, luck only comes in at around 2%. All major backgammon tournaments count doubling dice points and not games won or lost. In other words- you might win nine games, but if you lose a 36 pointer on the tenth game, you will have lost the match.
The key to the doubling dice is when to offer it and when to accept or refuse it. You should only ever offer it if you think that the other player will accept it. There is no point, most of the time, otherwise. To wait until they are obviously so down or weak and then give them a chance to limit the damage by losing a single game, rather than a double, is madness. So offer it when you can see that you have an advantage, but when you think that your opponent still holds out hope.
You should accept or refuse a double offered- purely on the merits of the game. Never accept on emotion or because the dice is low, so you think that you have little to lose. You must teach your opponent to be wary of offering it- by being prepared to take away there fun and end the game right there and then.
Doubling in backgammon is akin to raising in poker and this is when the game really comes alive. Some moderate players could by their sheer moves alone compete with a very strong player if there were no doubling cube. However when we add the cube then the better player can read a situation much deeper than the intermediate player and simply punish them more for losing.
An intermediate level play may tie an expert over say a 50 game run at 25 wins apiece. However when the cube is added then the expert may win say an average of 3.5pts for every win while the intermediate players wins 2.5pts. So if they are playing for say £1 per point then the expert wins £1 more than the other player for every game that they win and so are £25 ahead over these 50 games for a profit of £0.50 per game.