# Probability of one or all occurring ?

So I know the probability of multiple things happening involved multiplication but what about the probability of one or some or all events occurring? That seems like addition but then how could that be because wouldn’t in some cases the total meet or exceed one?

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• Some events overlap, ie they can both be true which is multiplication and some events are mutually exclusive, which means they add. But they all take place in a sample space that sums to 1. So really it is just a case of how that area of 1 in a Venn diagram is partitioned into events, and some things just can't fit into an area of 1, eg two mutually exclusive event with p = 0.9

• The easiest way to see the formula is with sets or a Venn diagram.

Suppose that out of 60 students, there are 10 females studying math, 10 males studying math, 20 females not studying math and 20 males not studying math.

We want to know the probability that a randomly selected student is female and/or studying math.

If we just add the 30 females and 20 math students, we have double counted the 10 females who are studying math. We can correct for that by subtracting 10 from our answer. Number of A or B = number of A + number of B - number of (A&B).

This math applies to probability too.

P(A or B) = P(A) + P(B) - P(A&B)

In the example above, the events were independent. 1/2 of students were female, 1/3 were studying math, and 1/6 were females studying math.

P(A or B) = P(A) + P(B) - P(A)*P(B)

But this version of the formula is only true when the events are independent.

When events are mutually exclusive, P(A&B) = 0 and therefore P(A or B) = P(A) + P(B)

In real life, events are often neither fully independent nor fully exclusive. For example, it could have been that only 7 female students chose to study math. In that case there's no helpful math; you can't tell from "1/2 female, 1/3 math" how big the overlap is.

Now let's consider 3 events. With some algebra or a Venn diagram you can come up with the following formula:

P(A or B or C) = P(A) + P(B) + P(C) - P(A&B) - P(A&C) - P(B&C) + P(A&B&C)

With even more events the formula is going to get really complicated really fast!

But there's a quick way to calculate the probability that one-or-more of the many events happens:

P(A or B or C or D...) = 1 - P(none of these)

If you roll 100 six-sided dice, what is the probability that you roll at least one 1?

For each die, the probability of NOT rolling a 1 is 5/6.

The dice are independent of each other, so the probability of them all not rolling a 1 is (5/6)^100.

Therefore the probability of rolling at least one 1 is (1 - (5/6)^100).

• You begin with a grain of truth. The probability of multiple events occurring does generally involve multiplication, but there is more to it than that. One big concern is the interdependence of those events.

Now, why are you confused about some or all events occurring? Those are both cases of multiple events. As for the probability of one event, that might be a bit simpler, but of course we would have to know something about that event.

• The probability of multiple things happening is known as the "intersection".

Pr(A intersecting B) = Pr(A) * Pr(B)

The probability of at least one event happening is known as the "union".

Pr(A in union with B) = Pr(A) + Pr(B) - Pr(A intersecting B)

It's useful to look at a Venn Diagram to properly understand where these formulas came from. You can google the words "Venn Diagram Probability" and there'll be many videos and images to help you understand this concept.