is a way to describe the likelihood
of something happening. You usually encounter probability
often without even realizing it. For example, when the weather report says that there is a 60% chance of rain today, that is an expression
of probability. And if someone says that you have a 50-50 chance of guessing a coin toss - that too, is an expression
of probability. While these are everyday occurrences that deal with probability, when we talk about probability
mathematically, we usually write probabilities either in fraction
or decimal form.
In general, we say that the probability
of something happening is the ratio
of the number of ways that thing can happen to the total number of ways for all things to happen. The thing we want to happen is usually called the event. So we will need to know the number of ways for the event to happen and the total number of ways for all events to happen. In a simpler form,
For example, let’s think about rolling a die (this is singular for dice). A die has six sides and a number from 1 to 6 appears on each side. We usually assume that we have a “fair die” meaning that each number has an equal chance of occurring.
Let’s talk about the probability
of rolling a 4. In this case, rolling a 4 is the event
we are interested in. There is only one way for this to happen, so 1 is the numerator of the ratio. Because there are six possibilities, 6 will be the denominator of the ratio. We usually express the probability
of rolling a 4 as:
The six possibilities that we use as the denominator of the ratio
are usually referred to as the sample space
. The sample space
is where you list all the possible outcomes for an experiment. Our experiment
was to roll the die one time. Sometimes listing the sample space
is very helpful in knowing how many outcomes there are in an experiment. Other times, like with rolling a die, you can just think about how many ways something can happen. And still other times, there may be so many possibilities that you don’t want to list them and you can’t just think about them easily. We won’t worry about those situations here, but be aware of all the ways you can deal with a sample space
(it’s simple and you don’t have to list it, it’s helpful to list all the possibilities, it’s too complicated to list every possibility).
It is interesting to note that the probability
of rolling any number 1 - 6 will always be
since all numbers have an equal chance of happening. It is also important to note that if you add all the probabilities together, you will get 1.
When you add all the probabilities associated with all the events of an experiment, you will get one. This is an important rule to remember when working with probabilities.
Knowing that all the probabilities associated with an experiment
equal one can save some time in working some problems.
What if a problem asked you to find the probability
of NOT rolling a 3? That means you would be looking for the probability
of rolling a 1, 2, 4, 5, or 6. You would need to find all five of those individual probabilities and add them up. But an easier way is to work with the complement. The complement
is a way of finding the probability
of an event NOT happening. So the probability
of NOT getting a 3 is found by finding the probability
of 3 and then subtracting from 1.
Another important rule deals with the type of numbers that are acceptable as a probability
answer. Probabilities can only take on values from 0 to 1. Keep in mind that 0 and 1 are acceptable values for a probability
answer. Mathematically this is represented as
For example, if we go back to our die problem, the probability
of rolling a 7 is zero because you can never roll a 7 with just one die.
that you are used a computer to access this lesson is 1 because the only way to see these lessons is on-line or by printing the web page. Either way, it is certain that you used a computer to access this lesson.
P(use a computer to access this lesson) = 1
If an event is neither certain nor impossible, then its probability
should be somewhere between 0 and 1. If you perform a computation for a probability
and your answer is negative or larger than 1, then your answer is incorrect. This will be useful in later lessons as you perform more complicated computations.
In addition to the die examples already discussed there are several other common types of problems that you will come across when working with probability.
- Marbles in a Box
Suppose there are 3 red marbles, 6 blue marbles, and 7 green marbles in a box. The probability
of pulling out a red marble is 3/16. There are 3 red marbles which gives us our numerator of 3 and there are 16 total marbles which gives us our denominator.
In order to solve problems involving cards, you should learn some basic facts about a deck of cards if you are not familiar with cards.
- There are 52 cards in a deck.
- The 52 cards are broken up into 4 suites (spades, clubs, diamonds, hearts).
- Each suite has 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10. Jack, Queen, King).
- There are 4 of each type of card in the deck (4 aces, 3 10’s, etc.)
- Two suites are red (diamonds and hearts).
- Two suites are black (spades and clubs).
So a simple probability
problem that is very common would be to ask what is the probability
of pulling out a 6. This probability
would be 4/52 which could then be simplified to 1/13. Notice there are 4 six’s in a deck out of a total of 52 cards which is how we get 4/52. You should always simplify your final answer when expressing a probability.
We have looked at some examples that deal with rolling one die. However it is very common to see problems that involve rolling two dice.
When rolling two dice, there are 36 possibilities. It is usually helpful to consider a first die and a second die to keep the two distinct. The possibilities listed below are ordered pairs indicating the number of the first die and then the number on the second die. To find the sum, simply add the two numbers.
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
So if we want to find the probability
of rolling a sum of 8, we need to find all the possible ways to roll an 8 (there are 5) out of the total possibilities which is 36.
Therefore, P(rolling sum of 8) = 5/36