I took a probability class in college and am currently going through a free MIT probability class on edX.org. Both classes never really showed why you can't always tell if 2 events are independent on a Venn diagram. In the full classroom lecture for the edX class on youtube, the professor explains that independence is not the same thing as disjoint events and then he hints at the fact that you can't use a Venn diagram to tell if 2 events are independent. He then goes on to show that 2 disjoint events are dependent using a Venn diagram. He then blurred through the case when 2 events are joint. He didn't do anything wrong, it was just confusing!

Anyhow, he was right, if the 2 events are disjoint, they are not independent because if one event occurs, then it is impossible for the other event to occur. So the knowledge of one event occurring, necessarily excludes the other event. This makes the 2 events dependent on each other.

To sum this up, in the case of disjoint events, you can tell 2 events are not independent, but if the events have an intersection (joint) they may or may not be independent. Therefore a Venn diagram can't indicate independence and in the case that 2 events are joint, it must either be given in the problem that the events are independent or reasoning and a mathematical test must be used to determine it.

An example of 2 events that are joint, but are independent, is the event that a card chosen from a deck of playing cards is a 4 and the conditional event that the card is a 4 of clubs given that you chose a club. The probability of a card being a 4 is 1/13 and the probability that the card is a 4 given that you chose a club is still 1/13. This comparison against the non-conditional event and the conditional event is the mathematical test to determine if 2 events are independent. The probabilities being the same confirms the 2 events are in fact independent.

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