Probability Calculator

Probability is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Basic Probability:

The basic probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This assumes that all outcomes are equally likely.

For example, the probability of rolling a 6 on a fair die is 1/6, because there is 1 favorable outcome (rolling a 6) out of 6 possible outcomes.

Conditional Probability:

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(B|A), which reads as "the probability of B given A."

For example, if we know that a student is a senior, what is the probability that they are taking calculus?

Union of Events:

The union of events A and B, denoted as A ∪ B, is the event that either A or B (or both) occurs. The probability of the union is calculated using the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

For example, what is the probability of drawing a heart or a face card from a standard deck of cards?

Intersection of Events:

The intersection of events A and B, denoted as A ∩ B, is the event that both A and B occur. For independent events, P(A ∩ B) = P(A) × P(B). For dependent events, P(A ∩ B) = P(A) × P(B|A).

For example, what is the probability of drawing two aces in a row from a standard deck of cards without replacement?

Independent vs. Dependent Events:

Two events are independent if the occurrence of one event does not affect the probability of the other event. Otherwise, they are dependent.

For example, rolling a die twice produces independent events, because the outcome of the first roll does not affect the outcome of the second roll.

Drawing cards from a deck without replacement produces dependent events, because the outcome of the first draw affects the composition of the deck for the second draw.