The difference between mutually exclusive and independent events is: a mutually exclusive event can simply be defined as a situation when two events cannot occur at same time whereas independent event occurs when one event remains unaffected by the occurrence of the other event.
Suppose two events have a non-zero chance of occurring. Then if the two events are mutually exclusive, they can not be independent. If two events are independent, they cannot be mutually exclusive.
One example is a disease and its vaccine. Suppose you have a 100% effective vaccine such that you cannot contract the disease if you were vaccinated. Then being vaccinated and contracting the disease are mutually exclusive and they are dependent. Event that can't occur simultaneously are mutually exclusive.
Independent events are those events whose occurrence is not dependent on any other event. For example, if we flip a coin in the air and get the outcome as Head, then again if we flip the coin but this time we get the outcome as Tail. In both cases, the occurrence of both events is independent of each other.
If events are disjoint then they must be not independent, i.e. they must be dependent events. Why is that? Recall: If A and B are disjoint then they cannot happen together. In other words, A and B being disjoint events implies that if event A occurs then B does not occur and vice versa.
Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.
Events A and B are independent if: knowing whether A occured does not change the probability of B. Mathematically, can say in two equivalent ways: P(B|A) = P(B) P(A and B) = P(B ∩ A) = P(B) × P(A).
Mutually Exclusive: can't happen at the same time. Examples: Turning left and turning right are Mutually Exclusive (you can't do both at the same time) Tossing a coin: Heads and Tails are Mutually Exclusive.
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0.
In probability, we say two events are independent if knowing one event occurred doesn't change the probability of the other event.
You can tell if two random variables are independent by looking at their individual probabilities. If those probabilities don't change when the events meet, then those variables are independent. Another way of saying this is that if the two variables are correlated, then they are not independent.
When we calculate probabilities involving one event AND another event occurring, we multiply their probabilities. In some cases, the first event happening impacts the probability of the second event. We call these independent events. ...