courses › Business Stats & Decisions

Probability Basics

level 21 course

Complements, 'both', 'at least one', and when you can't just multiply.

In your head, jot if needed · no calculator why?

Opens at level 21.

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A probability says how likely something is, from 0% (never) to 100% (always). It describes what happens across many similar cases, not a promise about one customer.

The chance something doesn't happen is 100% minus the chance it does. For two independent events, where one tells you nothing about the other, the chance both happen is the two chances multiplied. And "at least one" is best worked backward: 1 minus the chance of none.

Techniques

Multiply for both

Two independent events, and you want the chance both happen.

Write each chance as a decimal: 90% is 0.9.
Multiply them. The result is never bigger than either chance.
Multiply only when the events are independent.

worked example

Example: Two independent suppliers deliver on time 80% and 90% of the time. What is the chance both are on time?

0.8 × 0.9 = 0.72 = 72%.

Answer: 72%

At least one: 1 minus none

The chance of at least one success over several independent tries.

A miss on one try: 100% minus the chance of success.
Multiply the misses together: the chance of none at all.
At least one is 1 minus that.

worked example

Example: A food truck sends 3 independent catering bids, each with a 20% chance of winning. What is the chance of winning at least one?

Losing one: 0.8.
Losing all 3: 0.8 × 0.8 × 0.8 = 0.512.
At least one: 1 − 0.512 = 0.488 = 48.8%.

Answer: 48.8%

Ask what links them

Before multiplying chances, when one cause could hit both.

Ask what the two share: an industry, a port, a flood zone, an investor.
A shared risk can push the true chance of both well above the product.
If one event always brings the other, the chance of both equals the smaller chance.

worked example

Example: Two stores in the same flood zone each have a 10% chance of flood damage this year. If any flood always damages both, what is the chance both are damaged?

Multiplying would give 10% × 10% = 1%.
But one flood hits both, so both are damaged whenever either is.
So both: 10%, ten times the multiplied figure.

Answer: 10%

Tips by skill

TipThe chance it doesn't happen: It either happens or it doesn't: subtract the chance from 100%.
TipBoth happen: Independent events: multiply the chances as decimals, then write the result as a percent.
TipAt least one: Multiply the chances of a miss to get the chance of none, then subtract that from 100%.
TipCan you just multiply?: Ask whether one cause could hit both. If it could, the multiplied chance may be far too low, so don't trust it.

Watch out for

Repeating the given chance when the question asks for the other side: 35% canceling means 65% staying.
Adding chances for "both" or "at least one". Three 50% calls don't make 150%; no chance passes 100%.
Multiplying out of habit when a shared risk links the events. That can badly understate the chance that everything goes wrong at once.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

The chance it doesn't happen par 9 s not tried yet
worked example
A shipment has a 9% chance of arriving late. What is the chance it doesn't arrive late?

Answer: 91%
1. It either happens or it doesn't: 100% − 9% = 91%.
Both happen par 16 s not tried yet
worked example
Two machines each run a full shift without a breakdown 90% of the time, independently of each other. What is the chance both run a full shift without a breakdown?

Answer: 81%
1. For independent events, multiply: 0.9 × 0.9 = 0.81, which is 81%.
At least one par 19 s not tried yet
worked example
You submit two independent bids, each with a 30% chance of winning. What is the chance of winning at least one?

Answer: 51%
1. Chance of winning none = 0.7² = 0.49.
2. At least one = 1 − 0.49 = 0.51, which is 51%.
Can you just multiply? par 17 s not tried yet
worked example
Two clients funded by the same investor each have a 5% chance of losing their funding this year. Is 0.25% a safe estimate that both lose funding?
1. Yes — probabilities always multiply
2. No — you must add the two chances
3. No — a shared risk can make both happen together
4. Yes — just multiply the two chances
Answer: No — a shared risk can make both happen together
1. Multiplying 5% × 5% = 0.25% assumes the two are independent.
2. If the investor pulls back, both can lose funding at once, so the chance that both lose funding can be much higher than 0.25%.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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