How the Formula Works

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Goal: How much traffic or budget gives you a good chance of a conversion?

Suppose your conversion rate is p (e.g. 5%), and you want to know:
How many independent clicks (n) do I need for at least a given probability (q) of at least one conversion?

Step 1: Probability of no conversions

Each visit (or click) has a chance p of converting. That means the chance of not converting in one visit is 1-p.

For n independent visits, the probability no one converts is:

(Probability of 0 conversions after n visits) = (1-p)^n

Step 2: Probability of at least one conversion

We want the probability of at least 1 conversion in n visits.
This is 1 minus the probability of no conversions:

Probability (at least 1 conversion) = 1 - (1-p)^n

Let’s call this probability q.

Step 3: Solve for n

You want the probability above to be at least q. Rearranging the formula:

1 - (1 - p)n = q

We solve for n:

  1. Subtract 1 from both sides:
    (1 - p)n = 1 - q
  2. Take the natural logarithm ln() of both sides:
    ln((1-p)^n) = ln(1-q)
  3. Use log rules: ln(a^b) = b*ln(a)
    n * ln(1-p) = ln(1-q)
  4. Solve for n:
    n = ln(1-q) / ln(1-p)
Example: For a 5% conversion rate and 90% desired probability:
n = ln(1-0.90) / ln(1-0.05) ≈ ln(0.1)/ln(0.95) ≈ -2.3026 / -0.05129 ≈ 44.9
So, you need at least 45 visits to have a 90% chance of at least one conversion!
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