P03 · Lesson 1 of 10 · 11 min · 4 hands-on tasks
Thinking in Probabilities
Business is a game of chances, not certainties. See why small numbers bounce around, how chances combine, and why "it'll probably be fine" needs a number.
🎥 Watch it (2 min)
🎬 Situation
Nina runs Sprout Box, an online shop that delivers houseplants. Every day she makes calls she can't be sure about: Will this new plant sell? Will orders arrive safely? Is a good day luck, or a real trend?
She can't remove uncertainty. But she can learn to measure it.
🎮 Play
Visitors arrive one by one. Each has a 3% chance of buying. Press the buttons and watch the conversion rate as visitors pile up.
Nothing yet. Press a button to run some visitors.
Notice how wild the line is at the start. With 20 visitors, one lucky sale moves the rate by 5 points. With 2,000 visitors, the same sale barely moves it.
Chances combine. One damaged plant in 50 sounds rare. But Nina ships a lot of orders.
More likely than not, someone gets a damaged plant today. At this scale it's not bad luck, it's a process to fix.
🏷️ Name it
| Idea | Rule | Sprout Box |
|---|---|---|
| Probability | How often, in the long run | 3% of visitors buy |
| Both A and B (independent) | multiply: p(A) × p(B) | On time 90% × undamaged 95% = 85.5% |
| A or B (can't both happen) | add: p(A) + p(B) | Buys a plant 2% + buys a pot only 1% = 3% |
| Not A | 1 − p(A) | 97% of visitors don't buy |
| At least once in n tries | 1 − (1 − p)ⁿ | 64% chance of a damaged order among 50 |
| Law of large numbers | More tries → closer to the true rate | 20 visitors bounce; 2,000 settle |
🖍️ Like you're five: you have a bag with 3 red marbles and 97 blue ones. Grab one with your eyes closed: it's almost always blue. But grab one every day for a year and you'll get red about 11 times. Rare things happen all the time when you try often enough.
Real life: airlines, hospitals and factories think this way: a tiny chance of failure per step, multiplied over millions of steps, means failures are certain somewhere, so they build checklists and backups. And investors know a startup's first good week means little: they want to see the rate hold for months.
🔨 Do it: 4 tasks
0 / 4 doneTask 1 of 4 · From counts to a chance
Last month 12,000 people visited Sprout Box and 360 bought something. What is the chance a visitor buys?
Task 2 of 4 · Both have to happen
An order arrives on time 90% of the time. The plant arrives undamaged 95% of the time. The two don't affect each other.
What's the chance an order is both on time and undamaged?
Task 3 of 4 · At least once
Each order has a 2% chance of arriving damaged. Nina ships 50 orders today.
What's the chance at least one arrives damaged? (Round to the nearest whole percent.)
Task 4 of 4 · Small numbers
Nina's new plant page had 20 visitors yesterday and 2 bought. Her usual rate is 3%. What should she conclude?
🧩 Case: The great first week
⭐ VIPNina launches a new "Office Jungle" bundle. In its first 3 days it gets 40 visitors and 4 sales: a 10% conversion rate, more than three times her usual 3%. Her supplier needs an order today: buy stock for 200 bundles, or 50?
What does she order?
What would you do? VIP members make the call, see what happens next across 2 decisions, and get the expert debrief.
🔁 Remember it
A probability of 0.25 means...
A coin landed heads 5 times in a row. The next flip is...
Each of 3 independent steps succeeds 80% of the time. All three succeed...
A probability is how often something happens in the long run. Small samples bounce wildly. Both A and B (independent): multiply. At least once in n tries: 1 − (1 − p)^n.
📅 Use it Monday
Find one plan at work that depends on several things all going right. Put a rough probability on each step and multiply them. Then ask: is the overall chance as high as everyone assumes?
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