🧭 Compute the probability of 200 simultaneous requests for any test generator on the IN‑V‑BAT‑AI website. Can the system scale to 1,000,000 students within a 10‑minute usage interval? The answer is absolutely yes.

Because each generator is delivered as static HTML, static JavaScript, static CSS, and static PNG assets, modern browsers automatically cache these files on every student’s smartphone or tablet. This dramatically reduces server load and allows the platform to scale to extremely high levels of concurrent usage.

Let’s do a clean back‑of‑the‑envelope, the way you’d sketch it on a whiteboard.

🔧 1. Simple concurrency model

Assume:

𝑁 = number of students who will load a given fraction test generator during a class

Each student loads that fraction test generator once during a window of length 𝑇 seconds

Each HTTP request is “active” on the server for 𝑑 seconds (for a static file this is tiny; let’s take 𝑑 = 0.2 seconds as a conservative upper bound)

Requests are spread roughly uniformly over the window

Then the probability that a given student is “in flight” (request active) at a random instant is:

𝑝 = 𝑑 / 𝑇

The number of simultaneous requests 𝑋 at any instant is approximately:

𝑋 ∼ Binomial (𝑁,𝑝), 𝐸 [X] = 𝑁𝑝 = 𝑁 ⋅ 𝑑 / 𝑇

🧠 2. Plug in realistic classroom‑style numbers

Take a 10‑minute window or 600 seconds for a fraction test generator:

𝑇 = 600  seconds , 𝑑 = 0.2  seconds

Because the probability " 𝑝 " that any random student is loading the fraction test generator at the exact same instant is extremely small, the system experiences virtually no contention. In practice, this means web pages load instantly with no noticeable delay.

𝑝 = 0.2 / 600 = 1 / 3000 ≈ 0.00033

Case A: 𝑁 = 100,000 students using that fraction test generator in that 10‑minute window

𝐸 [𝑋] = 100,000 ⋅ 1 / 3000 ≈ 33.3  the Expected average or mean value of simultaneous requests or concurrent request

Expected average or mean value of concurrency ≈ 33

200 simultaneous requests is ~ 6 × 33 = 198 students opening the same fraction test generator at the same time. 6 x the mean (33), which is already extremely unlikely.

Using a Poisson approximation with 𝜆 = 33.3 ,

𝑃 (𝑋 ≥ 200 ) ≈ 0 ( astronomically small )

Case B: 𝑁 = 1,000,000 students in that same 10‑minute window

𝐸 [𝑋] = 1,000,000 ⋅ 1/ 3000 ≈ 333.3

Expected concurrency ≈ 333

Now 200 simultaneous requests is below the mean, so:

𝑃 (𝑋 ≥ 200 ) ≈ 1 ( almost certain )

But this assumes something very strong:

All 1,000,000 students are in the same 10‑minute window, hitting the same fraction test generator.

✅ What “a 10‑minute window for a fraction test generator” means

It does not mean the fraction test generator itself runs for 10 minutes.

It means this:

A 10‑minute window = the time period during which students are likely to load that fraction test generator.

Example:

A teacher opens your Fraction Pie Generator at 10:00 AM

Students begin using it between 10:00 and 10:10 AM

During that 10‑minute period, all students will load the same static HTML/JS/CSS/PNG files

So the “window” is simply the usage interval.

Why this matters for concurrency math:

Students do not all click at the exact same second

Their requests are spread across several minutes

Each request lasts only 0.1–0.2 seconds

So the chance of 200 requests landing in the same 0.2‑second slice is extremely small unless you have hundreds of thousands of students using the same fraction test generator at the same moment.