Introduction to Probability

Dr. Math FAQ || Classic Problems || Formulas || Search Dr. Math || Dr. Math Home

For a review of concepts, see Permutations and Combinations.

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.

* Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen.

Likewise, when dealing with infinities, a probability of 1 doesn't guarantee the event: when choosing a random number between 1 and 10, what is the probability that you'll choose a number other than 5.0724? It's 1.

#### One event, all outcomes equally likely

Suppose we have a jar with 4 red marbles and 6 blue marbles, and we want to find the probability of drawing a red marble at random. In this case we know that all outcomes are equally likely: any individual marble has the same chance of being drawn.

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

number of favorable outcomes
---------------------------------------
total number of possible outcomes

• What's a favorable outcome? In our example, where we want to find the probability of drawing a red marble at random, our favorable outcome is drawing a red marble.

• What's the total number of possible outcomes? The total number of possible outcomes forms a set called a sample space (see note). In our problem, the sample space consists of all ten marbles in the jar, because we are equally likely to draw any one of them.

Using our basic probability fraction, we see that the probability of drawing a red marble at random is:
 number of red marbles 4 --------------------------- = --- total marbles in jar 10

Since 4/10 reduces to 2/5, the probability of drawing a red marble where all outcomes are equally likely is 2/5. Expressed as a decimal, 4/10 = .4; as a percent, 4/10 = 40/100 = 40%.

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:

1. (you draw a blue marble and then I draw a blue marble)
2. (you draw a blue marble and then I draw a red marble)
3. (you draw a red marble and then I draw a blue marble)
4. (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.

#### On the Web:

How does a basic knowledge of probability help us understand
what's happening in the real world?