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Probability Calculator

Calculate various probabilities including P(A or B), P(A and B), conditional probability P(A|B), and probabilities for independent or mutually exclusive events.

Probability Calculator

Calculate various event probabilities. Enter probabilities as decimals (e.g., 0.5 for 50%).

Probability of Complement (P(A'))

Calculates the probability of event A NOT occurring.

Formula: P(A') = 1 - P(A)

Understanding Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. A probability is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility and 1 indicates certainty.

Complement of an Event: P(A')

The complement of an event A, denoted A', is the event that A does not occur. The rule of complements states that the probability of an event occurring is 1 minus the probability that it does not occur.

Formula: P(A') = 1 - P(A)

Example: If the probability of rain P(A) is 0.3, the probability of no rain P(A') is 1 - 0.3 = 0.7.

Union of Events: P(A or B)

The probability of the union of two events A and B is the probability that either A or B (or both) will happen. The general addition rule is used to find this.

Formula: P(A or B) = P(A) + P(B) - P(A and B)

We subtract P(A and B) because this intersection area is counted twice when we add P(A) and P(B) together.

Conditional Probability: P(A|B)

This is the probability of event A occurring, given that event B has already occurred. It narrows down the sample space to only the outcomes where B happens.

Formula: P(A|B) = P(A and B) / P(B)

Example: The probability of drawing a King of Hearts is 1/52. The probability of drawing a heart is 13/52. Given that you drew a heart, the probability of it being the King is (1/52) / (13/52) = 1/13.

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin twice. The outcome of the first flip does not influence the outcome of the second.

For independent events, the probability of both occurring is: P(A and 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, you cannot turn both left and right at an intersection.

For mutually exclusive events, the probability of their intersection is zero: P(A and B) = 0. This simplifies the addition rule to: P(A or B) = P(A) + P(B).

Understanding Your Chances: A Guide to Probability

Probability is the measure of the likelihood that an event will occur. It's a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. I designed this calculator to handle several fundamental probability scenarios, helping you make sense of the odds.

This tool can calculate probabilities for single events, the union of two events (A or B), conditional probabilities (A given B), and special cases for independent and mutually exclusive events.

Key Probability Formulas Explained

1. Complement of an Event: P(A')

This is the probability that event A does NOT occur. It's the simplest calculation.

P(A') = 1 - P(A)

2. Union of Events: P(A or B)

This is the probability that event A or event B (or both) will happen. The general addition rule is:

P(A or B) = P(A) + P(B) - P(A and B)

We subtract P(A and B) because this intersection is counted twice when we add P(A) and P(B).

3. Conditional Probability: P(A|B)

This is the probability of event A happening, given that event B has already occurred.

P(A|B) = P(A and B) / P(B)

4. Special Cases

  • Independent Events: The occurrence of one event does not affect the other. Here, P(A and B) = P(A) * P(B).
  • Mutually Exclusive Events: The events cannot happen at the same time. Here, P(A and B) = 0, so the addition rule simplifies to P(A or B) = P(A) + P(B).

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