When all outcomes are equally likely, the probability of an event is the fraction favorable outcomes / possible outcomes, and it always falls between 0 (impossible) and 1 (certain). In an urn with 3 red balls and 2 blue ones, drawing a red ball is 3/5. Drawing two reds in a row without putting the first one back is 3/10, and you will see why below. If you start from counted data instead, see relative frequency with fractions.
| The problem says | Operation | Example |
|---|---|---|
| A single event | Favorable over possible | Rolling a 5 on a die: 1/6 |
| "Not," "none," "at least one" | 1 minus the probability of the opposite | Not rolling a 6: 1 − P(6) = 5/6 |
| "Or" | Add, and subtract the overlap | Even or a multiple of 3 on a die: 3/6 + 2/6 − 1/6 = 2/3 |
| "And," one after another | Multiply | Heads and a 6: P(heads) × P(6) = 1/12 |
The keyword in the problem decides the operation, and the rest is careful counting and simplifying. The rule of dividing favorable outcomes by possible outcomes only works when the outcomes are equally likely.
With two dice there are 36 equally likely ordered pairs. A sum of 7 comes from six of them (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1), so P(sum of 7) = 6/36 = 1/6.
If you put the first ball back, the urn again has 3 red out of 5, and the probability is 3/5 × 3/5 = 9/25. Since 0.3 is less than 0.36, not replacing the ball lowers the probability: fewer reds are left for the second draw. For the product you can use the multiply fractions calculator.
The Complement of an Event
The complement of A happens when A does not, and its probability is 1 − P(A). Not rolling a 6 on a die is 1 − 1/6 = 5/6, which matches counting the five favorable outcomes out of six.
The shortcut helps when the opposite is easier to count. For “at least one 6 with two dice,” the opposite is “no 6”: each die misses with probability 5/6, and since they are independent, 5/6 × 5/6 = 25/36. So P(at least one 6) = 1 − 25/36 = 11/36. Counted on a 6 × 6 grid, that is the last row and the last column: 6 + 6 − 1 = 11 cells out of 36.
Worked Example: Two Reds Without Replacement
An urn has 3 red balls and 2 blue balls. You draw two balls, one after the other, without putting the first one back. What is the probability that both are red?
- First ball. There are 3 reds out of 5 balls, so 3/5.
- Second ball. If the first was red, 4 balls remain and 2 of them are red, so 2/4.
- Multiply. Both things must happen, so 3/5 × 2/4 = 6/20.
- Simplify. The greatest common factor (GCF) of 6 and 20 is 2, and 6/20 = 3/10.
The probability is 3/10, which is 0.3 or 30%.
Check. There are 5 × 4 = 20 ordered pairs of balls, and 3 × 2 = 6 of them are two reds. That gives 6/20 again.
Events with “Or”
If the events cannot happen at the same time, you add them. A 1 or a 2 on a die is 2 favorable outcomes out of 6, that is, 2/6.
If they can happen at the same time, you have to subtract the overlap, because it was counted twice. In a standard 52-card deck there are 13 hearts and 4 kings, and the king of hearts is both:
P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13.
Counting directly, there are 13 hearts plus 3 kings from other suits, which makes 16 favorable cards. To add probabilities with different denominators, use the add fractions calculator.
Events with “And”
If the result of one event does not change the probabilities of the other (they are independent), you multiply. With a coin and a die, heads and a 6 is 1/2 × 1/6 = 1/12.
If the first result changes what is left, as when you draw without replacement, the second fraction is computed from what remains, as in the urn example. In any other case, ask yourself the same question: does the second probability depend on what happened before?
Probability and Observed Frequency
If you toss a coin 20 times and get 13 heads, 13/20 = 0.65 is an observed frequency. The probability for a fair coin is still 1/2. Over long runs the frequency tends to get close to the probability, but in a short run it is normal for it to drift away. The guide on relative frequency with fractions explains how to calculate it from real data.
To turn a probability into a percent, the fraction to percent calculator shows the full value and a decimal approximation; the simplify fractions calculator reduces the result.
Practice
- A bag has 4 green chips and 6 yellow ones. What is the probability of drawing a green one?
- With a die, what is the probability of rolling an odd number or a 4?
- You toss two coins. What is the probability of two tails?
- A bag has 5 strawberry candies and 3 lemon ones. You draw two without replacement. What is the probability that both are strawberry?
Show the answers
- 4/10 = 2/5
- There are 3 odd numbers and 4 is not odd, so they do not overlap. You get 3/6 + 1/6 = 4/6 = 2/3
- 1/2 × 1/2 = 1/4
- 5/8 × 4/7 = 20/56 = 5/14
More exercises with answers are in worked fraction exercises.
Sources
- English Wikipedia, article Probability, sections “Interpretations” and “Mathematical treatment,” on probability as a number between 0 and 1, the ratio of desired outcomes to all outcomes, and the complement as 1 − P(A) (accessed October 17, 2026).