Probability Calculator
Results
Show Formula and Steps
What Is Probability?
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the toss of a fair coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
Conditional Probability & Bayes’ Theorem Explained
Conditional Probability is the probability of one event occurring in the presence of a second event. The conditional probability of event A given that event B has occurred is written as P(A|B). The formula is: P(A|B) = P(A ∩ B) / P(B). For example, what is the probability that a selected card is an Ace, given that we know it is a Spade? P(Ace|Spade) would be the probability of the card being the Ace of Spades (1/52) divided by the probability of the card being a Spade (13/52), which equals 1/13.
Bayes' Theorem is a fundamental theorem in probability theory that describes how to update the probability of a hypothesis based on new evidence. It is particularly useful in medical diagnostics and machine learning. The formula is: P(H|E) = [P(E|H) * P(H)] / P(E). It allows us to calculate the "posterior" probability P(H|E) using the "prior" probability P(H), the "likelihood" P(E|H), and the probability of the evidence P(E).
Independent vs Mutually Exclusive Events
Understanding the difference between these two concepts is crucial in probability:
- Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. The classic example is flipping a coin twice; the result of the first flip has no impact on the second. For independent events A and B, the probability of both happening is
P(A ∩ B) = P(A) × P(B). - Mutually Exclusive Events: Two events are mutually exclusive (or disjoint) if they cannot both occur at the same time. For example, when rolling a single six-sided die, the outcomes of rolling a 2 and a 3 are mutually exclusive. For mutually exclusive events A and B, the probability of either happening is
P(A ∪ B) = P(A) + P(B). The probability of both happening, P(A ∩ B), is 0.
Permutations & Combinations — When to Use Each
Both permutations and combinations are used to count the number of ways a set of items can be selected. The key difference is whether the order of selection matters.
- Permutations (Order Matters): Use permutations when the arrangement or sequence of the selected items is important. For example, determining the first, second, and third place winners in a race. The formula is
P(n, k) = n! / (n - k)!. - Combinations (Order Does NOT Matter): Use combinations when the order of selection is irrelevant. For example, choosing a committee of 3 people from a group of 10. The specific people on the committee matter, but the order in which they were chosen does not. The formula is
C(n, k) = n! / (k! * (n - k)!).
Frequently Asked Questions
Q: What is the range of a probability value?
A: A probability value is always a number between 0 and 1, inclusive. 0 represents an impossible event, and 1 represents a certain event.
Q: What is the complement of an event?
A: The complement of an event A, denoted A', is the event that A does not occur. The probability of the complement is P(A') = 1 - P(A).
Q: How does this calculator handle fractions and decimals?
A: You can enter probabilities as decimals (e.g., 0.5), fractions (e.g., 1/2), or from outcome counts. The calculator will process the input and provide the result in multiple formats (fraction, decimal, and percentage) for your convenience.
This calculator provides educational probability calculations. Verify results for professional or critical use.
