Probability Calculator

Probability is the measure of the likelihood that an event will occur, quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Our Probability Calculator helps you compute the probabilities of two independent events, including their complements, intersection, union, exclusive or, and neither occurring. Simply enter the probabilities of events A and B, and the calculator instantly displays all derived probabilities.

How to Use the Probability Calculator

1. Enter P(A) - the probability of event A occurring (a value between 0 and 1).
2. Enter P(B) - the probability of event B occurring (a value between 0 and 1).
3. View all results instantly, including complements, intersection, union, exclusive or, and neither.

Understanding the Probability Calculations

Complement: P(A') and P(B')

The complement of an event is the probability that the event does NOT occur. It is calculated as: $$P(A') = 1 - P(A)$$. For example, if there is a 65% chance of rain (P(A) = 0.65), then the probability of no rain is P(A') = 1 - 0.65 = 0.35 or 35%.

Intersection: P(A ∩ B)

The intersection of two events is the probability that BOTH events occur simultaneously. For independent events, this is calculated as: $$P(A \cap B) = P(A) \times P(B)$$. For example, the probability of rolling a 6 on two separate dice rolls is (1/6) × (1/6) = 1/36 or approximately 0.0278.

Union: P(A ∪ B)

The union of two events is the probability that AT LEAST ONE of the events occurs. It is calculated using the inclusion-exclusion principle: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$. For example, the probability of rolling an even number or a multiple of 3 on a single die roll is 3/6 + 2/6 - 1/6 = 4/6 or 0.6667.

Exclusive OR: P(A △ B)

The exclusive or (XOR) of two events is the probability that exactly ONE of the events occurs, but NOT both. It is calculated as: $$P(A \oplus B) = P(A) + P(B) - 2 \times P(A \cap B)$$. This excludes the scenario where both events happen simultaneously.

Neither: P((A ∪ B)')

The probability that NEITHER event occurs is the complement of the union: $$P((A \cup B)') = 1 - P(A \cup B)$$. This represents the probability that no event from the set of A or B takes place.

Applications of Probability

• Statistics and Research: Calculate significance levels, confidence intervals, and hypothesis testing.
• Finance: Assess investment risks, portfolio probabilities, and insurance premiums.
• Gaming: Determine odds in card games, dice games, lotteries, and sports betting.
• Quality Control: Evaluate defect rates and reliability in manufacturing processes.
• Weather Forecasting: Interpret probability of precipitation and other weather events.

Frequently Asked Questions