Probability Calculator
Use this calculator to determine the probability of single or multiple independent events. Input the number of favorable outcomes and the total possible outcomes for each event.
Optional: For calculating combined probabilities of two independent events.
Results:
Probability of Event A (P(A)):
Probability of Event B (P(B)):
Probability of A AND B (P(A ∩ B)):
Probability of A OR B (P(A ∪ B)):
Understanding Probability
Probability is a fundamental concept in mathematics that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Often, probabilities are also presented as percentages (0% to 100%).
Basic Probability Formula
The most basic way to calculate the probability of a single event is:
P(Event) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
- Favorable Outcomes: These are the specific outcomes you are interested in.
- Total Possible Outcomes: This is the complete set of all possible results of an experiment or situation.
Example: If you roll a standard six-sided die, the total possible outcomes are {1, 2, 3, 4, 5, 6}, so there are 6 total outcomes. If you want to find the probability of rolling a '3', there is only one favorable outcome (the number 3). So, P(rolling a 3) = 1/6.
Independent Events
Two events are considered independent if the occurrence of one does not affect the probability of the other occurring. For example, flipping a coin twice are independent events; the result of the first flip does not influence the second.
Probability of A AND B (Intersection)
When you want to find the probability that two independent events, A and B, both occur, you multiply their individual probabilities:
P(A AND B) = P(A) × P(B)
Example: What is the probability of rolling a '3' on a die (Event A) AND flipping 'Heads' on a coin (Event B)?
- P(A) = 1/6
- P(B) = 1/2
- P(A AND B) = (1/6) × (1/2) = 1/12
Probability of A OR B (Union)
When you want to find the probability that either event A occurs OR event B occurs (or both), for independent events, the formula is:
P(A OR B) = P(A) + P(B) – P(A AND B)
This formula accounts for the possibility that both events might occur, preventing double-counting. Since P(A AND B) = P(A) × P(B) for independent events, it can also be written as:
P(A OR B) = P(A) + P(B) – (P(A) × P(B))
Example: What is the probability of rolling an 'even number' (Event A) OR rolling a 'number greater than 4' (Event B) on a single die roll? (Note: These are not independent in the context of a single roll, but for demonstration of the formula with two distinct events, let's consider two separate dice rolls for simplicity, or two distinct events like rolling an even number on one die and flipping heads on a coin).
Let's use a different example for clarity with independent events:
What is the probability of rolling a '3' on a die (Event A) OR flipping 'Heads' on a coin (Event B)?
- P(A) = 1/6
- P(B) = 1/2
- P(A AND B) = 1/12 (as calculated above)
- P(A OR B) = (1/6) + (1/2) – (1/12) = (2/12) + (6/12) – (1/12) = 7/12
Using the Calculator
Input the number of favorable outcomes and total possible outcomes for Event A. If you wish to calculate combined probabilities, also input the values for Event B. The calculator will then provide the individual probabilities and the combined probabilities for "AND" and "OR" scenarios, assuming the events are independent.