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In this lesson, you will use the concept of probability to discuss common situational probabilities and study some helpful counting techniques that are commonly used when calculating probability.
Probability is the study of how likely something is to happen; more specifically, probability helps us decide how likely it is that a certain outcome will follow from an event. Think of probability as a function that takes one possible outcome from an event and returns a number between 0 and 1.
If the probability of that particular outcome takes the value 0 then the event is impossible, meaning it can never happen. If the probability of that particular outcome takes the value 1, then the event is guaranteed to occur, meaning it must happen. Other possible outcomes have probabilities that lie somewhere between these extremes.
Probability is a numerical value ranging from 0, meaning impossible, to 1, meaning guaranteed.
As stated earlier, the probability function assigns a number between 0 and 1 to any particular outcome following from an event. This gives a good intuition for determining probability, but how do we actually determine the exact number? For any event, we consider the possible outcomes, and we determine which of these outcomes are favorable. If an outcome is favorable, it is not necessarily desirable. It just means that it is the outcome for which we are evaluating probability. For example, if we are assessing the probability that it will rain, then rain is a favorable outcome, regardless of whether we would be happy if it rained.
First, let’s consider a special situation in probability: a situation in which each outcome is equally likely, such as rolling a fair die and considering the probability of rolling any particular number, or drawing a card from a standard deck and considering the probability of drawing a particular card. If we let E stand for a favorable outcome, and P(E) stand for the probability that we get a favorable outcome, then we have the following formula:
This formula says that the probability of a favorable outcome is the ratio of the number of favorable outcomes to total number of possible outcomes. The total number of possible outcomes is called the sample space.
Suppose we flip two coins simultaneously. What is the probability that both coins will land with their tail sides facing up?
The correct choice is B. In this example, E = both coins land with their tail sides facing up. To use the formula, we must determine the ratio of favorable outcomes to total outcomes. If we use H to represent heads and T to represent tails, we can see there are four total outcomes: HH, HT, TH, TT. Clearly, only one of these four outcomes provides the favorable outcome of two tails (TT), and thus 
.
Let’s consider another interesting point in computing probabilities. We can reason in the following way: Either a favorable outcome will happen or it won’t. Thus E or (not E) must occur. Because the probability of a guaranteed event is equal to 1, we get P(E or (not E)) = 1. Furthermore, because P(E or (not E)) = P(E) + P(not E), we get P(not E) = 1 – P(E).
Sometimes it is not clear how to count the favorable outcomes and the total outcomes. In this case, it is usually easier to count the outcomes for (not E). At times like this, we have to use this equation to compute P(E) using P(not E).
Suppose we let E represent the outcome “a die lands on a number less than or equal to 4.” Then 
. The probability that a die lands on either 5 or 6 is 
, which is exactly what we would expect from calculating 
explicitly.
As with all mathematics, the numerical value of a probability has real-world implications. Keeping the formula for probability in mind, we can easily interpret the numerical value of a probability measure. Clearly, the probability is zero only when there are no favorable outcomes for an event. Similarly, when each and every outcome is favorable, the probability is equal to one. In the example above, we found that
. Realistically, this means if we flip two coins 100 times, we should expect 25 of those flips to return two tails. Or, if we flip the two coins four times, we should expect to get two tails once. Of course, these are rough estimates. The more trials we run, the closer the result will be to 25%.
When choosing one card at random from a deck of cards, what is the probability of choosing a face card (i.e., a jack, queen, or king)? There are 52 cards in a deck and 12 face cards total.
The correct choice is B. In this example, E = “choosing a face card from a single deck.” We are only choosing one card, so the number of favorable outcomes is equal to the number of face cards in one deck. There are four suits, each with three face cards, which means that there are 12 cards that we can pick to achieve a favorable outcome. There are 52 cards in a deck, so we have 52 total outcomes, therefore 
.
Important Tidbit
This hierarchy of the possibility of events determines the payoff when gambling. The less likely an event is to occur, the higher the payoff for the gambler when it does. In poker, the probability of being dealt a hand containing a pair (a hand with any two cards alike) is much greater than the probability of being dealt a royal flush (a hand containing an ace, a king, a queen, a jack, and a ten, all from the same suit). Because a royal flush is so unlikely by comparison, it beats a pair (and everything else for that matter). Therefore, the royal flush holder gets the big payoff.
The formula for calculating probability is quite simple. It tells us that probability really just boils down to counting two things: the number of favorable outcomes for an event and the number of total outcomes. This definition of probability is only true when every single outcome is as equally likely as any other. If we are to have an accurate count, we need to make sure we do not forget any of these outcomes. So far, all the counting we have needed to perform has been fairly easy. As one would expect, however, some events require much more sophisticated counting techniques to determine their probability.
The most important of these techniques is the fundamental counting principle. This principle states that the total number of possible outcomes following from a series of events is determined by multiplying all the ways that each individual event can occur.
Suppose that a license plate consists of two letters followed by four digits. The plates use only capital letters, and only the numbers 0 through 9 (these letters and numbers can repeat; for example, a license plate might feature more than one 3 or more than one A). What is the total number of unique license plates possible?
The correct answer is D. We have six places that have to be filled on the license plate.
We can think of each place as an individual event. In each blank, we write the number of ways that each place can be filled. Assuming we use only capital letters and numbers 0 through 9, we have 26 choices for the letter slots and 10 choices for the number slots.
Using the fundamental counting principle, the total number of license plates possible is 
. Clearly, the counting principle is a very useful tool.
Spotting when to use the counting principle can be a tremendous help. Counting the ways that a complicated event can occur can be tricky. If it isn’t clear how to count the different outcomes, try to break the event into a series of events or choices. Once you have a series of events from which to choose, use the counting principle.
Here is another example of the counting principle. Suppose we have 7 books to put on a bookshelf and we want to know how many different arrangements are possible. We can look at this one event of arranging books on a shelf as seven individual events or choices.
We have 7 books and thus 7 choices for the first slot.
Because only six books are left after the first slot is filled, we have six choices for the second slot.
Continuing this way and multiplying, we find that there are
ways of arranging seven books on the shelf. Recall that the ! operation is called the factorial.
and is read “n factorial.”
When computing factorials it is important to remember the following facts:
The factorial is defined only for nonnegative numbers (0, 1, 2, 3, . . . ).
So you should never have to compute –7! or –3!. These computations do not make sense. If they arise, check your work again to see if you made a mistake.
For notational purposes, 0! = 1.