The probability of A is 0.70, the probability of B is 0.80,...

The probability of $A$ is $0.70$, the probability of $B$ is $0.80$, and the probability of both is $0.50$. What is the conditional probability of $A$, given $B$ ? Are $A$ and $B$ independent in a probability sense?

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Key Concepts

Conditional Probability

Conditional probability quantifies the likelihood of an event occurring given that another event has already occurred. It is defined mathematically as the probability of the intersection of two events divided by the probability of the given event, provided that the probability of the given event is non-zero.

Event Independence

Independence between two events implies that the occurrence of one event does not affect the probability of the occurrence of the other. In mathematical terms, two events are independent if and only if the probability of their intersection equals the product of their individual probabilities, which is also equivalent to the conditional probability of one event given the other being equal to its original probability.

Intersection of Events

The intersection of events refers to the event where both events occur simultaneously. It is a fundamental concept in probability that helps in computing joint probabilities and is integral to the definitions of conditional probability and independence.

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