I have a forecast that says the wind is going to veer by 30º overnight.
Suppose if I jibe now and the wind does veer 30º by tomorrow morning I will gain. But If I jibe now and the wind does not veer 30º by tomorrow morning then i will lose.
What do I need to know to decide if I should jibe?
We need numbers, or estimates of numbers, on all three factors involved.
P = probability of the wind veering as forecasted.
Gain = how much you gain (in hours or miles) if you jibe and the wind shifts as forecasted
Risk = how much you lose (in the same units) if you jibe and the wind does not shift as forecasted.
We can figure the Risk and Gain numbers from our polars.
We are left with computing a take point probability P for the forecast. In other words, we calculate how big does P have to be so that the chances of gaining are higher than the chances of losing.
P x Gain ≥ (1-P) x Risk.
In words, the probability of the veer (P) times the Gain from the veer must be greater than the probability of no veer (1-P) times the Risk. If the probability of yes is 70% or 0.7, then the probability of no has to be 30% or 0.3.
Now you can rearrange the terms to get
P ≥ Risk / (Risk + Gain),
which is our working guideline, and we can make a table to solve it by inspection. The units can be anything, miles, minutes, hours.
On the other hand, you face a more ambitious maneuver to catch up. You figure you will gain 2 mi but would lose 8 mi if there is no veer. Then you have P = 8/10 = 80%. The forecast has to have an 80% chance of being right, which makes you study the weather maps very carefully, look for ship reports, read the Forecast Discussions, etc. Do what ever you can to add confidence to the forecast.
If the winds aloft were changing rapidly, and the present surface analysis did not match what you actually saw in pressure and wind, then you probably can't believe this is 80% likely, and have to pass, or maybe try 3 hr and then jibe back to wait another 3 hr to get the next map. In other words, do half of what you want.
Needless to say, one can take that formula off of the boat and think about it in various social or economic settings. The decoration on the table is in honor of our lunch and break table at work, where it rules about 12.5% of our working day.