What Probability Means
“Probability is 'favorable ÷ total', a ratio between 0 and 1.”
The formula
P(A) = a / nHow to read it: when there are n equally likely outcomes and a of them make event A happen, the probability of A is a/n
- n
- — the number of all possible outcomes (each equally likely)
- a
- — how many of them make event A happen
- a / n
- — the probability of event A
The hook
Probability nails 'how often will it happen' down to a single number between 0 and 1.
In plain words
Count the equally likely outcomes, then take the share of them that are the ones you're hoping for.
The intuition
A die lands on 6 faces equally. The evens (2, 4, 6) are 3 of those 6 → half. Probability is 'your share of all the slots'. 0 means never, 1 means certain.
How it's built
The denominator is the total number of outcomes, the numerator is the number of favorable ones. But each outcome must be equally likely (fair) for this ratio to be the probability.
Example
Probability of rolling an even number on one die: of 6 outcomes, the evens are 2, 4, 6 → 3 → 3/6 = 1/2.
Common mistake
A probability of 1/2 doesn't mean you're guaranteed one hit in two tries. Probability is what the ratio settles toward over the long run, not a promise on any single try.
Where it's used
Weather forecasts, insurance, games, medical tests — the basis of every judgment that handles an uncertain world with numbers.
Where it came from
In the 1600s Pascal and Fermat traded letters on 'how to split the pot if a bet is stopped partway', lifting probability into mathematics.
Prerequisites
Quick check
Rolling one die, what is the probability of a multiple of 3 (a 3 or a 6)?
- 1/6
- 1/3✓
- 1/2
- 2/3
Practice
From balls numbered 1 to 10, you draw one. What is the probability of a multiple of 3 (3, 6, 9)? Give it as a decimal.
Answer: 0.3
- Multiples of 3: 3, 6, 9 → 3 of them
- 3 of 10 outcomes → 3/10 = 0.3
Takeaway: Probability = favorable outcomes ÷ total outcomes.
Rolling one die, what is the probability of a prime (2, 3, or 5)?
Answer: 2
- Primes on a die: 2, 3, 5 → 3 of them
- 3/6 = 1/2
Takeaway: First count the favorable outcomes carefully.
From cards numbered 1 to 20, you draw one. Find the probability of a multiple of 4.
Answer: undefined
- Multiples of 4: 4, 8, 12, 16, 20 → 5 of them
- 5/20 = 1/4
Takeaway: Find the numerator and denominator, then reduce the fraction.