The study of probability helps us figure out the likelihood of something
happening. For instance, when you roll a pair of dice, you might ask how
likely you are to roll a seven. In math, we call the "something happening"
an "event."

The probability of the occurrence of an event can be expressed as a fraction
or a decimal from 0 to 1. Events that are unlikely will have a probability
near 0, and events that are likely to happen have probabilities near 1.*

In any probability problem, it is very important to identify all the
different outcomes that could occur. For instance, in the question about
the dice, you must figure out all the different ways the dice could land,
and all the different ways you could roll a seven.

To find a basic probability with all outcomes equally likely, we use a fraction:

Suppose we number the marbles 1 to 10. What is the probability of picking out
number 5?

Well, there is only one number 5 marble, and there are still 10 marbles in the jar, so the answer is 1 marble (favorable outcome) divided by 10 marbles (size of sample space) = 1/10 or 10 percent.

**Note:**
What is a **sample space**?
The sample space is a set consisting of all the possible outcomes of an
event (like drawing a marble from a jar, or picking a card from a deck).
The number of different ways you can choose something from the sample space is the total number of possible outcomes.

Because each probability is a fraction of the sample space, the sum of the
probabilities of all the possible outcomes equals one. The probability of the occurrence of an event is always one minus the probability that it doesn't occur.

For example, if the probability of picking a red marble from a jar that
contains 4 red marbles and 6 blue marbles is 4/10 or 2/5, then the probability of
*not* picking a red marble is equal to 1 - 4/10 = 6/10 or 3/5, which is
also the probability of picking a blue marble. Given the only
two events that are possible in this example (picking a red marble or
picking a blue marble), if you don't do the first, then you must do the
second. That is, given this example, the probability of picking a red marble plus the probability of picking a blue marble will equal 1 (or 100 percent).

then - Two
events, all outcomes equally likely

Now let's suppose we have two events: first you draw 1 marble from the 10,
and then I draw another marble from the nine that remain. What is the
probability that you will draw a *blue* one first? What is the
probability that I will draw a *red* one second
(given that you have already drawn a blue one)?
Again, we'll use our fraction. When you draw the first marble, there are 10
marbles in the jar of which 6 are blue, so your probability of drawing a
blue one is 6/10 (60 percent) or 3/5.

After you draw it's my turn, but now the size of our sample space has
changed because there are only 9 marbles left; 4 of them are red, so the
probability that I'll draw a red marble (again, assuming that you
have already drawn a blue one) is 4/9.

and - Two
events, second outcome dependent upon the first

We just calculated the probability for *two* events whose outcomes were equally
likely: in the first, you drew a blue marble; in the second, I drew a red
one *after* you had drawn.
But suppose we want to know the probability of your drawing a blue marble
**and** my drawing a red one?
This may seem like the same question, but it is *not* the same because
we now have more than one event. Here are the possibilities that make up
the sample space:

- (you draw a blue marble
and then I draw a blue marble)
- (you draw a blue marble
and then I draw a red marble)
- (you draw a red marble
and then I draw a blue marble)
- (you draw a red marble
and then I draw a red marble)

These are the only four possibilities - but they are not all equally likely.
When we have an event *made up of two separate events* with the word and, where the outcome of the second event is dependent on the
outcome of the first, we multiply the individual probabilities to get the answer.

- The probability of example (b) above, is:
your probability of drawing a blue marble (3/5) multiplied by my probability
of drawing a red marble (4/9): 3/5 x 4/9 = 12/45 or, reduced, 4/15.

- How about the probability of example (a)?
We've already calculated the probability of your drawing a blue marble;
it's 3/5. How about the probability of my drawing a blue marble too?

Well, after you draw a blue, there are 9 marbles left and 5 of them are
blue, so for me the probability will be 5/9. Multiply 3/5 times 5/9 and you
get 3/9 or, reduced, 1/3.

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