Value of a Bird in the Hand in Terms of Birds in Bushes

Is a bird in the hand really worth two in the bush? A GEICO commercial, where an antiques appraiser values a sculpture of a bird in the hand as worth "two in the bush" got me thinking about that expression. Maybe a bird in the hand is really worth 3 or 5 birds in the bush. I'm no bird hunter, but I am an amateur economist, and I know to find the value of a bird in the hand in terms of birds in bushes, we can break it down mathematically. The expression "a bird in the hand is worth two in the bush" can be mathematically expressed by the following equation: valueOfBirdInHand = 2*valueOfBirdInBush

But this is inadequate. To fully flesh out the economic wisdom contained in this colloquialism, we need several variables other than the number of birds dwelling in the bush. These variables are as follows: 1) The probability of catching a bird, given that it is in the bush.
2) The fixed costs involved with going bird-catching in the first place. By fixed costs, meaning the same amount of these costs must be incurred regardless of the bird-yield.

3) The variable costs that must be incurred per bird that is caught. Variable meaning that these costs increase per caught bird.

So adding in these variables, by my reckoning, the equation for the value of an attempt to hunt birds in the bush is:
totalValue= P(catching bird)*numBirds*unitValue - (numBirds*unitVariableCost) - fixedCost
with P(catching bird) denoting the probability of catching a bird given that it is in the bush. Unit value gives a measurement of value to a caught bird, for example the prevailing market rate for that particular bird.
To explain this perhaps perplexing equation, P(catching bird)*numBirds*unitValue gives the total expected value, meaning you might catch 50% of 10 birds valued at $20 each for $100 total yield, or 25% of 12 birds valued at $5 each for a $15 yield. NumBirds*unitVariableCost gets you your total variable cost. Subtract that out along with your fixed cost and you have the expected net value of your hunting trip. It's like an expected profit.

So to find the value of a bird in the hand, set totalValue to the value of a single bird, and solve for numBirds in the following equation:
P(catching bird)*numBirds*unitValue - (numBirds*unitVariableCost) - fixedCost = unitValue

Solving,

factor out numBirds and move fixedCost to other side,
numBirds*(P(catching bird)*unitValue - unitVariableCost)=unitValue+fixedCost
divide both sides by (P(catching bird)*unitValue-unitVariableCost)) and we have our answer:
numBirds=(unitValue + fixedCost)/(P(catching bird)*unitValue-unitVariableCost)
This equation answers the question of how many birds in the bush equal the value of one bird in the hand! For example, let's say birds were worth $10 each, the probability of catching a bird in a bush was 0.50, and there were zero fixed or variable costs involved with bird catching. In this case:
numBirds = ($10 + 0)/(0.50*$10 - 0)
numBirds = 1/0.50
numBirds = 2
Thus in that situation a bird in the hand is worth two in the bush, just like the expression tells us.
But what if there were fixed and variable costs involved with hunting for birds? What if you needed to pay a $5 fee to go bird hunting, and it cost you $1.00 per bird to get it ready for market? Then:

numBirds = ($10+$5)/(0.5*$10-$1)

numBirds = 15/4
numBirds = 3.75
So in this case a bird in the hand would be worth 3.75 birds in the bush.
Now what if, keeping the costs the same, the probability of catching a bird in a bush decreased to 0.25?:
numBirds=($10+$5)/(0.25*$10-1)
numBirds=15/1.5
numBirds=10
In this case a bird in the hand would be worth 10 birds in the bush.
So after all these mathematical gyrations you might think I'm being silly just like the GEICO commercial. Well, yes I am being silly but there are also valuable economic lessons to be learned here. You can replace "bird" with any other thing of value, and the simple economic model I have assembled here, as well as the wisdom of the expression it was based on, would be just as valid. The old saying examines one of the great conflicts in economic life: the sure thing vs. speculative gain. For more on that topic check out my article on the game show "Deal or No Deal".
The moral of the story is: to evaluate a decision, look at the probability of a favorable outcome, look at the variable costs, and look at the fixed costs.