Probability Calculator
Calculate probability for single events, complements, unions, intersections, and conditional probability. See also Combinations Calculator and Percentage Calculator.
How to Calculate Probability
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). For a single event, divide the number of favorable outcomes by the total number of possible outcomes. For combined events, use the addition rule (union), multiplication rule (intersection), or Bayes' theorem (conditional).
Probability Formulas
Single Event: P(A) = Favorable / Total
Complement: P(A') = 1 − P(A)
Union: P(A∪B) = P(A) + P(B) − P(A∩B)
Intersection: P(A∩B) = P(A) × P(B|A)
Independent: P(A∩B) = P(A) × P(B)
Conditional: P(A|B) = P(A∩B) / P(B)
Example Calculation
A bag has 3 red, 4 blue, and 5 green marbles (12 total).
P(red) = 3/12 = 1/4 = 0.25 = 25%
P(not red) = 1 − 0.25 = 0.75 = 75%
Odds for red = 3:9 = 1:3
Odds against red = 9:3 = 3:1
Probability Rules Reference
| Rule | Formula | When to Use |
|---|---|---|
| Addition | P(A∪B) = P(A)+P(B)−P(A∩B) | Either A or B occurs |
| Multiplication | P(A∩B) = P(A)×P(B|A) | Both A and B occur |
| Complement | P(A') = 1 − P(A) | Event does NOT occur |
| Conditional | P(A|B) = P(A∩B)/P(B) | A given B occurred |
Solved Examples
Example 1: Drawing Cards
What is the probability of drawing a red face card (Jack, Queen, or King of hearts or diamonds) from a standard 52-card deck?
Total face cards = 12 (3 per suit x 4 suits)
Red face cards = 6 (3 hearts + 3 diamonds)
P(red face card) = 6/52 = 3/26 = 0.1154
Answer: The probability is 6/52 = 3/26, approximately 11.54%. There are 6 favorable outcomes (J, Q, K of hearts and diamonds) out of 52 total equally likely outcomes.
Example 2: Independent Events - Weather and Traffic
The probability of rain tomorrow is 0.30. The probability of a traffic jam (independent of weather) is 0.20. What is the probability of both rain and a traffic jam?
P(Rain) = 0.30
P(Traffic) = 0.20
Since events are independent:
P(Rain AND Traffic) = P(Rain) x P(Traffic) = 0.30 x 0.20 = 0.06
Answer: The probability of both events occurring is 0.06 or 6%. For independent events, multiply their individual probabilities.
Example 3: Complementary Probability
A quality inspector finds that 3% of products are defective. In a batch of 5 independent items, what is the probability that at least one is defective?
P(defective) = 0.03, P(not defective) = 0.97
P(all 5 good) = 0.97^5 = 0.8587
P(at least one defective) = 1 - P(all good)
= 1 - 0.8587 = 0.1413
Answer: There is a 14.13% chance of finding at least one defective item. Using the complement rule (1 minus the probability of the opposite event) is much easier than calculating every possible combination of defectives.
Practice Questions
Question 1
Two dice are rolled. What is the probability that the sum is 7?
Answer: Total outcomes = 36. Combinations summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 ways. P(sum=7) = 6/36 = 1/6 = 0.1667 or about 16.67%.
Question 2
A bag contains 4 red, 3 blue, and 5 green marbles. If you draw 2 marbles without replacement, what is the probability both are red?
Answer: P(1st red) = 4/12. P(2nd red | 1st red) = 3/11. P(both red) = (4/12)(3/11) = 12/132 = 1/11 = 0.0909 or about 9.09%.
Question 3
In a class of 30 students, 18 study math and 12 study physics. If 6 study both, what is the probability a randomly selected student studies math or physics?
Answer: Using the addition rule: P(M or P) = P(M) + P(P) - P(M and P) = 18/30 + 12/30 - 6/30 = 24/30 = 4/5 = 0.80 or 80%.
Common Mistakes
Adding probabilities when you should multiply
For events occurring together (AND), multiply probabilities. For either/or situations (OR), add probabilities (subtracting overlap). P(A and B) = P(A) x P(B) for independent events. P(A or B) = P(A) + P(B) - P(A and B).
Assuming independence when events are dependent
Drawing cards without replacement creates dependent events. The probability of the second draw depends on what happened first. P(2nd ace | 1st ace) = 3/51, not 4/52.
Confusing "at least one" with exactly one
For "at least one" probability, use the complement: P(at least 1) = 1 - P(none). Trying to enumerate all cases (exactly 1, exactly 2, ...) is error-prone and unnecessarily complex.
Key Takeaways
- Probability ranges from 0 (impossible) to 1 (certain). P(event) = favorable outcomes / total outcomes for equally likely events.
- Multiplication rule: P(A and B) = P(A) x P(B|A). For independent events, P(B|A) = P(B).
- Addition rule: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, P(A and B) = 0.
- Complement rule: P(not A) = 1 - P(A). Use this for "at least one" problems.
- Conditional probability P(A|B) = P(A and B) / P(B) measures probability of A given that B has occurred.
Frequently Asked Questions
What is the difference between odds and probability?
Probability is the ratio of favorable outcomes to total outcomes (e.g., 1/4). Odds compare favorable to unfavorable outcomes (e.g., 1:3). They express the same information differently.
What does independent events mean?
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A∩B) = P(A) × P(B). Example: flipping a coin and rolling a die.
What is conditional probability?
Conditional probability P(A|B) is the probability of A occurring given that B has already occurred. It is calculated as P(A∩B) / P(B).
Can probability be greater than 1?
No. Probability always ranges from 0 (impossible) to 1 (certain). If a calculation gives a value outside this range, check your inputs.
What is the complement rule?
The complement rule states that P(not A) = 1 − P(A). It is useful when calculating the probability of an event NOT happening is easier than calculating it directly.