What if you’re trying to figure out the chance that neither of two events happens?
Still, think of a coin toss and a die roll at the same time. And what’s the probability that you don’t get heads and you don’t roll a six? That’s the “probability of not A and not B.” It sounds simple, but most people skip the subtlety that comes with independent versus overlapping events.
What Is Probability of Not A and Not B
When we talk about “not A” we mean the complement of event A—whatever happens that isn’t A.
Also, similarly, “not B” is the complement of B. The phrase “not A and not B” asks: *what’s the probability that both complements occur simultaneously?
In plain language, it’s the chance that event A doesn’t happen and event B doesn’t happen, all at once.
If you’re dealing with a single random experiment that can produce multiple outcomes, you’re looking for the probability of the subset of outcomes that lie outside both A and B Easy to understand, harder to ignore..
The math behind it
For any two events A and B in the same probability space:
[ P(\text{not }A \text{ and not }B) = 1 - P(A \cup B) ]
That’s the complement rule in a nutshell.
If A and B are disjoint (they can’t happen together), the formula simplifies to:
[ P(\text{not }A \text{ and not }B)=1-P(A)-P(B) ]
If they can overlap, you have to subtract the overlap once:
[ P(\text{not }A \text{ and not }B)=1-P(A)-P(B)+P(A\cap B) ]
Why It Matters / Why People Care
You’ll run into this calculation in everyday life:
- Risk assessment: estimating the chance that neither of two hazards occurs.
- Quality control: probability that a product fails neither in test A nor test B.
- Game design: figuring out odds that a player doesn’t hit any of two undesirable outcomes.
If you ignore the overlap between A and B, you’ll overstate the risk (or understate the safety).
In finance, overlooking the intersection of two market events can lead to disastrous portfolio decisions.
How It Works (Step by Step)
1. Identify the events and their probabilities
First, write down P(A) and P(B).
If you’re not given them directly, calculate them from the sample space.
2. Check if A and B are independent, mutually exclusive, or overlapping
- Independent: the outcome of one doesn’t affect the other.
(P(A\cap B)=P(A)P(B)). - Mutually exclusive: they can’t both happen.
(P(A\cap B)=0). - Overlapping: they can both happen.
You’ll need (P(A\cap B)) explicitly or via conditional probability.
3. Apply the right formula
- Independent:
(P(\text{not }A \text{ and not }B)= (1-P(A))(1-P(B))). - Mutually exclusive:
(1-P(A)-P(B)). - General case:
(1-P(A)-P(B)+P(A\cap B)).
4. Verify with a Venn diagram
Draw A and B, shade the overlap, and see what remains uncovered. That uncovered area is exactly what you’re after.
5. Double‑check edge cases
- If P(A)=0 or P(B)=0, the result should be 1 minus the other event’s probability.
- If both events are certain (P=1), the probability of not both is 0.
Common Mistakes / What Most People Get Wrong
- Forgetting the overlap: Many people just subtract P(A) and P(B) and forget to add back the intersection.
- Assuming independence blindly: If you see two dice rolls, you might think they’re independent, but if they’re part of the same experiment (e.g., a single roll of a die that can show “A” or “B”), independence doesn’t hold.
- Mixing up “and” vs “or”: The complement of “A or B” is “not A and not B,” but the complement of “A and B” is “not A or not B.”
- Overlooking sample space size: In small sample spaces, probabilities can be misleading if you don’t count outcomes correctly.
- Using percentages as probabilities without conversion: 30% is 0.30, not 30.
Practical Tips / What Actually Works
- Always write a quick Venn diagram first. It forces you to think about overlap.
- Use the complement rule: (P(\text{not }A \text{ and not }B)=1-P(A\cup B)). It’s reliable regardless of independence.
- When in doubt, compute (P(A\cap B)) directly. Even if it seems tedious, it clears up ambiguity.
- Check consistency with total probability: The sum of probabilities for all disjoint outcomes should equal 1.
- make use of software for complex cases: A simple script in Python or Excel can handle large sample spaces and multiple events.
- Remember the “short version”: For independent events, just multiply the complements.
[ P(\text{not }A \text{ and not }B)= (1-P(A))(1-P(B)) ]
FAQ
| Question | Answer |
|---|---|
| What if A and B are not independent? | Use the general formula: (1-P(A)-P(B)+P(A\cap B)). In real terms, |
| **Can I use the same approach for more than two events? ** | Yes, but you’ll need inclusion‑exclusion. |
| **How do I find (P(A\cap B)) if it’s not given?That said, for three events: (P(\text{not }A \text{ and not }B \text{ and not }C)=1-P(A)-P(B)-P(C)+P(A\cap B)+P(A\cap C)+P(B\cap C)-P(A\cap B\cap C)). ** | It turns a “not” question into a “yes” question, which is often easier to calculate. Otherwise, calculate directly from the sample space. |
| What if the events are mutually exclusive? | If you know (P(A |
| Why is the complement rule so powerful? | Then (P(A\cap B)=0) and the formula reduces to (1-P(A)-P(B)). |
The probability of not A and not B is more than a textbook trick; it’s a tool for making sense of uncertainty in real life. By keeping the complement rule front of mind, respecting overlap, and double‑checking with a quick diagram, you’ll avoid the most common pitfalls and get a clear, accurate answer every time.
