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how do you know if probability is dependent or independent ?

Here are some links that might be useful to you:

https://www.statisticshowto.com/probability-and-statistics/dependent-events-independent

https://www.statisticshowto.com/probability-and-statistics/dependent-events-independent
Dependent Events and Independent Events – Statistics How To
How to tell if an event is Dependent or Independent. Being able to tell the difference between a dependent and independent event is vitally important in solving probability questions. Why? Imagine a single event: winning the lotto. That depends upon you buying a ticket. So winning the lotto and buying a ticket are events.

https://www.khanacademy.org/math/ap-statistics/probability-ap/stats-conditional-probability/a/check-independence-conditional-probability

https://www.khanacademy.org/math/ap-statistics/probability-ap/stats-conditional-probability/a/check-independence-conditional-probability
Conditional probability and independence (article) | Khan Academy
In probability, we say two events are independent if knowing one event occurred doesn’t change the probability of the other event. For example, the probability that a fair coin shows heads after being flipped is 1 / 2 ‍ .

https://www.khanacademy.org / math / statistics-probability / probability-library / multiplication-rule-dependent / v / independent-events-1

https://www.khanacademy.org / math / statistics-probability / probability-library / multiplication-rule-dependent / v / independent-events-1
Independent & dependent probability (video) | Khan Academy
In shorthand code: Independent is when P (A|B)=P (A). In human words A is going to do whatever it does regardless of what B does. If one is causing or interfering with the other it’s dependent and you can spy that out by noticing that P (A|B) CHANGES from P (A). And so on.

https://brilliant.org/wiki/probability-independent-events

https://brilliant.org/wiki/probability-independent-events
Probability – Independent events | Brilliant Math & Science Wiki
In probability, two events are independent if the incidence of one event does not affect the probability of the other event. If the incidence of one event does affect the probability of the other event, then the events are dependent. There is a red 6-sided fair die and a blue 6-sided fair die. Both dice are rolled at the same time.

https://www.mathsisfun.com/data/probability-events-independent.html

https://www.mathsisfun.com/data/probability-events-independent.html
Probability: Independent Events – Math is Fun
So, for Independent Events: P (A and B) = P (A) × P (B) Probability of A and B equals the probability of A times the probability of B. Example: your boss (to be fair) randomly assigns everyone an extra 2 hours work on weekend evenings between 4 and midnight.

https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/independent-dependent-probability

https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/independent-dependent-probability
Analyzing event probability for independence – Khan Academy
So if you had the probability of B given A is equal to the probability of B. Same argument. That would mean they are independent. Or, if we said that the probability of A and B is equal to the probability of A times the probability of B then this also means they are independent. We know that this one is true. The of A and B is 1/6.

https://stats.libretexts.org/Bookshelves/Probability_Theory/Supplemental_Modules_(Probability)/Probability_and_Independence

https://stats.libretexts.org/Bookshelves/Probability_Theory/Supplemental_Modules_(Probability)/Probability_and_Independence
Probability and Independence – Statistics LibreTexts
7 janv. 2024Definition: Probability. We define probability of an event E to be to be. P(E) = number of simple events within E total number of possible outcomes. We have the following: P(E) P ( E) is always between 0 and 1. The sum of the probabilities of all simple events must be 1. P(E) + P(not E) = 1.

https://stats.libretexts.org/Courses/Fullerton_College/Math_120:__Introductory_Statistics_(Ikeda)/03:_Probability/3.05:_Independent_Events

https://stats.libretexts.org/Courses/Fullerton_College/Math_120:__Introductory_Statistics_(Ikeda)/03:_Probability/3.05:_Independent_Events
3.5: Independent Events – Statistics LibreTexts
14 juill. 2023Formula: Multiplication Rule. Multiplication Rule: If A and B are independent events, then P(A ∩ B) = P(A) ⋅ P(B) P ( A ∩ B) = P ( A) ⋅ P ( B). Be careful with this rule. You cannot just multiply probabilities to find an intersection unless you know they are independent. Also, do not confuse events with mutually exclusive …

https://math.libretexts.org/Courses/Fullerton_College/Math_100:_Liberal_Arts_Math_(Claassen_and_Ikeda)/06:_Probability/6.03:_Independent_Events_and_Conditional_Probabilities

https://math.libretexts.org/Courses/Fullerton_College/Math_100:_Liberal_Arts_Math_(Claassen_and_Ikeda)/06:_Probability/6.03:_Independent_Events_and_Conditional_Probabilities
6.3: Independent Events and Conditional Probabilities
14 janv. 2023Solution. Let A A = getting heads and B B = rolling 6. We know that P(A) = 1 2 P ( A) = 1 2 and P(B) = 1 6 P ( B) = 1 6. A A and B B are independent since the probability of rolling a 6 does not depend on the outcome of the coin toss. Using the multiplication rule, we get: P(A and B) = P(A) ⋅ P(B) = 1 2 ⋅ 1 6 = 1 12.

https://online.stat.psu.edu/stat800/lesson/how-do-we-check-independence

https://online.stat.psu.edu/stat800/lesson/how-do-we-check-independence
How do we check for independence? | STAT 800 – Statistics Online
Recall that two events are independent when neither event influences the other. That is, knowing that one event has already occurred does not influence the probability that the other event will occur. How can we check whether two events are independent using probabilities? There are three simple ways to check for independence:
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