Do independent events add to 1? I have been learning about independent events in statistics.
The textbook we have been given (pg. 12) states that "independent events do not add up to 1".
I'm really confused by this. When I learnt about tree diagrams, all the probabiliteis added to one, like in this example:Independent Events
So why do independent events add to one in that video, while the textbook states that they do not?
 A: Independent events do not necessarily have probabilities that sum to 1. All that is necessary for two events to be independent are $P(A \cap B) = P(A) P(B)$.
Perhaps you are thinking about a partition. We call a collection of events $A_1, \ldots, A_n$ a partition if $A_i \cap A_j = \emptyset$ for any $1 \leq i, j \leq n$ (that is, all the sets are disjoint) and $A_1 \cup \ldots \cup A_n = \Omega$, where $\Omega$ is the sample space. (In which case their probabilities do sum to 1.)
In my experience students confuse disjointedness and independence; that is, if they try to draw a Venn diagram representing independent events, they draw disjoint events instead. But disjoint events are not independent, because if $A$ and $B$ are disjoint and $A$ occurs, you automatically know that $B$ did not occur.
There's not much that you can say about events who's probabilities sum to one. They need not be independent, and they may not even be a partition. Probability allows for sets that are not empty but still have probability zero.
A: Two events $A$ and $B$ are defined to be independent if one of them happening doesn't interfere with the other. Mathematically this is expressed by
$$ P(A \cap B) = P(A)P(B) $$
To illustrate, let's say $A$ = It will rain tomorrow and $B$ = You pick a card from a deck and it's a King. It's obvious these events are independent and, clearly, you can have $P(A) + P(B) > 1$. So, no, independent events do not necessarily add up to $1$, but it may happen by coincidence.
