Can probabilities sometimes add to greater than one?
I read on my textbook that in a certain scenario, the sum of the probabilities is supposed to equal to one.
However, I read an example of an event on this site, and it says in some cases, independent events can add to greater than one. For example, if you add the probability of the chance of getting a red card from a pack of cards OR getting a tails on a coin flip, these events could be greater than one, and the sample space could therefore be greater than one.
Is this example, could the probabilities add to greater than one, and is it possible for the sample space to be greater than one? Or is this impossible?
$\endgroup$ 42 Answers
$\begingroup$Of course the sum of probabilities of events can be greater than $1$... however the probability of the union of events can never be greater than $1$. Reworded, $Pr(A)+Pr(B)$ is allowed to be greater than one but $Pr(A\cup B)$ is never greater than $1$.
As for the follow-up question "is it possible for the sample space to be greater than one" I assume you mean the probability of the sample space. No. No probability is ever allowed to be greater than one, ever.
Note: $Pr(A\cup B) = Pr(A)+Pr(B)\color{red}{-Pr(A\cap B)}$. The probability of a union of events is equal to the sum of the probabilities of the events only in the situation that the intersection of those events is impossible (i.e. $Pr(A\cap B)=0$)
$\endgroup$ 6 $\begingroup$Suppose we have three balls in a bag, one red (R), one blue (B), and one yellow (Y), well mixed, and you draw one from the bag.
The probability of drawing R is $1/3$. The probabilities of drawing B and drawing Y are the same. The probability of drawing R or B is $2/3$, as is the probabilities of drawing R or Y, and B or Y. The probability of drawing R, B, or Y is $1$. If you add all of these probabilities, you get$$\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + 1 = 4.$$
It's easy to get probabilities that sum to more than $1$. The question is, if they're summing to more than $1$, then why are you summing them? It doesn't represent anything meaningful in terms of what you're trying to model. In the above example, I can't think of any use I would have for the sum of the probability that I would draw R $(1/3$) and the probability that I would draw R, B, or Y ($1$) to make $4/3$. I don't see how this number makes anything about the above situation clearer.
There is a situation where adding two probabilities does tell you something: when you add mutually exclusive events. If it is impossible for two events to happen at the same time (or at least, the probability of them happening together is $0$), then adding the probabilities of these events will tell you the probability of one or the other happening. For example, because it's impossible to draw B and Y at the same time, the probability of drawing B or Y is $1/3 + 1/3$, the probability of drawing B plus the probability of drawing Y. Whereas, the probability of drawing R or (R, B, or Y) is not $4/3$, because the event of drawing R and the event of drawing R, B, or Y are definitely not mutually exclusive, because they can both be simultaneously satisfied by drawing R.
So, yes, it's possible to sum probabilities to more than $1$, but such numbers are not relevant, and are not even probabilities. It's impossible sum the probabilities of mutually exclusive events to be more than $1$, since the result is a relevant probability: the probability that one of the events will occur.
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