In mathematics and the calculus of probability, one can find a formula used more in gambling than anything else. In fact, remember that the calculus of probability was born to increase earnings in that field, around the 16th century, after the birth of the first bank (Monte dei Paschi di Siena). This is one of those cases where a discipline is born for practical reasons and then has a later theoretical generalization: for its grammarization we will need to wait for a 20th-century Russian. That formula has various names: mathematical expectation, expected value, and is used to calculate the average payoff of a game. Or, in the case of a company, of a strategy that has randomness, taking us into corporate game theory.
Let’s take a marketing campaign and schematize it this way from a probabilistic point of view:
- p= favorable cases/possible cases
- favorable cases = number of conversions, buyers
- possible cases = number of possible buyers, it usually costs a lot to know this value, you need extensive market research or demand forecasting. Let’s cut it short and identify it as the number of people who saw the advertisement go through (impressions).
- q= 1 – p
- V = win
- average purchase resulting from the advertisement (AOV)
- or selling price of the product
- B = bet
- cost per thousand (CPM)
- B comes from bet, in English
Mathematical formula: E= V * p + q * (-B)
why “E?” From the word expected in English.
let’s take this example
- p= 0.04% or conversion rate from impressions
- I thank @giuseppe.adveristing.2 for the true data
- q= 1-0,04 = 0,9996
- Win = 40€
- Bet = 6€/1000=0.006€
- cost per bet
- 1000 bets
With these data we have that E = 40 * 0.0004 + (-0.006) * 0.9996 = 0.4-0.00198=0.010€
For every 1000 bets we have 10€ revenue, but if we also introduce the cost of sales, i.e. 10€:
E* = (40-10) * 0,0004 + (-0,006) * 0,9996 = 0,3-0,00198=0,0060€
We have 6€ of revenue.
If the owner decided to make infinite bets, he would have a theoretical infinite net gain, but of course p does not turn out to be a constant, because sooner or later you saturate the market and this would be seen with p tending to 0 (and more, advertising data side), if it is tracked over time. So part of the game is figuring out when to stop playing. Kind of like choosing when to sow, which some grandparents know.
Assuming V the selling price decided by the owner, one might wonder if it turns out to be correct. In statistichese, whether the game turns out to be fair. Obviously not because he wants profit. But doing B/p one finds 15, greater than 40 (seen above). It just so happens that 15 also turns out to be the Customer Acquisition Cost, rediscovered in probabilistic terms. If doing B/p finds the value of V (V=B/p), the game would be fair. Unfortunately, many entrepreneurs discover, late in the game, that V< B/p, which means unfairness to themselves for the benefit of consumers. They may thus discover, again probabilistically, the negative cash flow, one of the main reasons for failure of SMEs.
As you may have guessed, data science, or statistics, can help with p, V (meaning AOV) and partially with B, to create more inherent ads, along with someone in the marketing department. If you are interested in learning more about how to do this with your company, we can start with a free call, and then continue with a statistical consultation.
As an exercise, the reader can check, if there is one, the relationship between E, ROI and ROAS.