To take the 3 points or not?

As the game continues to modernize, old adages become progressively less relevant and, if followed, can become destructive for a given team.
Such an adage that springs to mind is, "take the points", or, "keep the scoreboard ticking".
Let's look at the Mathematics behind the frequently visited question of whether to take the points or kick for touch.
I will outline the imgainary match situation, and then look at some basic math to give us some idea.
The Sharks are playing the Lions in Durban. No rain is forecast and it is fairly dry underfoot, there is little wind.
After 1 minute of play, the score is 0-0.
The Lions are given a penalty 30m away from the try-line and 20m in-field.
Let's look at the expected value of going for three points:
The Lion's goal-kicker has an 80% success rate from this position.
So the expected value of going for posts is 3x0.8 = 2.4 points
Now let's assume the goal kicker also has an 80% success rate with conversions and his average kick for touch ends up 10m from the try-line to form a lineout. His conversion therefore has an expected value of 2x0.8 = 1.6 points. Thus the act of scoring a try has an expected value of 5+1.6 = 6.6 points
We need to figure out how often the lions need to score a try, in order to justify kicking for touch.
Let P be the probability of scoring a try from a 10m lineout.
6.6 x P = 2.4
P = 0.36
So to justify turning down the 3 points, the lions must be able to score a try from this position more than 36% of the time. 
Obviously a lot of research would need to be done in order to understand how often the lions do in fact score from this position. My instinct tells me it would certainly be more than 36% of the time. 
There are many factors that would come into a real-life situation:
- As time goes on, it becomes less about expected value and more about match situation. For example if it is the 77th minute and the lions lead 36-30, we must take the kick here because the critical target is 38 points in this situation. (Critical target means the theoretical number of points needed to ensure victory.)- If they do take the points, we must remember we sacrifice a lot of territory as the opponent team gets the restart.- Many more that I can't be bothered thinking of.
However, when it is fairly early in the match and the score is 0-0, the equation becomes all about expected value because neither 3 points or 7 points is the critical target.
I believe in general that teams are taking the 3 points too often with the exception of the majority of the New Zealand teams.
I understand that this is in no way a comprehensive analysis, but it's a start to get us thinking.






 

Meanwhile the Lions have taken longer than the allocated minute whilst fishing around in their gear bag for a calculator and slide rule and the ref awards the Sharks a penalty try for time wasting...