The word probability means a number that represents how likely something is to happen.
If something is absolutely certain to happen, we says its probability is one. If something is impossible, we say its probability is zero. A number between zero and one means we don't know for sure what will happen, but the higher the number the more likely something is to happen. If something has a probability of 0.8, most people would say "it probably will happen".
Since we're using math to help us understand what will happen, it may help us to be able to write an equation like this:
Pr(X) = y
If this equation is true, we would say "the probability of X happening is y". For example, we can say:
X = A coin toss coming up heads
Pr(X) = 0.5
Since the letter X is standing for an event, not a number, and we don't want to get them confused, we'll use capital letters for events, and lower-case letters for numbers. The symbol Pr is represents a function that has some neat properties, but for now we'll just think of it as a short way of writing "the probability of X".
Since probabilities are always numbers between 0 and 1, we will often represent them as percentages or fractions. Percentages seem natural when someone says something like "I am 100% sure that my team will win." However, fractions make more sense for really computing probabilities, and it is very useful to really be able to compute probabilities to try to make good decisions, so we will use fractions here.
The basic rule for computing the probability of an event is simple, if the event is the kind where we are making a definite choice from a known set of choices based on randomness that is fair. Not everything thing is like that, but things like flipping a coin, rolling a single die, drawing a card, or drawing a ticket from a bag with your eyes closed are random and fair since each of the possibilities is equally likely. But the probability of things a little more complicated, like the sum of two dice, is not so easy---and that makes it fun!
Suppose we have a single die. It has six faces, and as far as we know each is equally likely to be face up if we roll the die. We can now use our basic rule for computing probability to compute the probability of each possible outcome. The basic rule is to take the number of outcomes the represent the event (called X) that you're trying to compute the probability of, and divide id by the total number of possible outcomes. So, for rolling a die:
Pr("gettting a one") = 1 face that has one dot / 6 faces = 1/6
Pr("gettting a two") = 1 face that has two dots / 6 faces = 1/6
Pr("gettting a three") = 1 face that has three dot / 6 faces = 1/6
Pr("gettting a four") = 1 face that has four dots / 6 faces = 1/6
Pr("gettting a five") = 1 face that has five dot / 6 faces = 1/6
Pr("gettting a six") = 1 face that has six dots / 6 faces = 1/6
So we have 6 possibilities, and each as a probability of 1/6. We know that 6 * (1/ 6) = 1. In fact, that is an example of a fundamental rule of probabilities:
The sum of probabilities of all possibilites is equal to one.
But you probably know that each face of a die is equally likely to land face-up, so so far we haven't done anyting useful. So let's ask the question: What is the probability of throwing two dice and having their sum equal seven when we add them together? Knowing that can help us win at a lot of games (such as Monopoly(trademark Parker Brothers)), so it's pretty valuable to know the answer. Suppose you are on St. Charles and another player has hotels on New York, St. James, and Tennesee (5, 7, and 8 spaces away). What is the probability that your next roll will force you to pay his high rent?
To find out the answer, we apply our basic rule, but now it is not so easy to know how many possible outcomes of TWO dice thrown together will equal seven. However, it is easy to know how many total possible outcomes there are: 6 times 6 = 36. One way to find out how many possible dice throws add up to seven is just to make a table of all possible dice throws that fills in their sum and count the number of sevens. Here is one:
***| 1 | 2 | 3 | 4 | 5 | 6
===========================
1 | 2 | 3 | 4 | 5 | 6 | 7
2 | 3 | 4 | 5 | 6 | 7 | 8
3 | 4 | 5 | 6 | 7 | 8 | 9
4 | 5 | 6 | 7 | 8 | 9 | 10
5 | 6 | 7 | 8 | 9 | 10| 11
6 | 7 | 8 | 9 | 10| 11| 12
The numbers along the top represent the value on the first die and the values on the left represent the values of the second die when you throw them together. The values inside the table reprsent the sum of the first and the second dice. (Note that the numbers range from 2 to 12 --- there's no way to throw two dice and get a number less than two!)
So if we could carefully, we can see that there are exactly 6 sevens in this table. So the probability of getting a seven if you throw two dice is 6 out of 36, or:
Pr(getting a 7) = 6 / 36 = 1 / 6.
(1 / 6 is a simple fraction than 6 / 36, but they are equal in value.)
Maybe that doesn't surprise you, but let's look at all the probabilities:
Pr(getting a 2) = 1 / 36
Pr(getting a 3) = 2 / 36
Pr(getting a 4) = 3 / 36
Pr(getting a 5) = 4 / 36
Pr(getting a 6) = 5 / 36
Pr(getting a 7) = 6 / 36
Pr(getting a 8) = 5 / 36
Pr(getting a 9) = 4 / 36
Pr(getting a 10)= 3 / 36
Pr(getting a 11)= 2 / 36
Pr(getting a 12)= 1 / 36
What do you think we would get if we added all of those probabilities together? They should equal 36 / 36, right? Check that they do.
Did you know that the probability of a getting a 5 is four times that of getting a two? Did you know that probability of getting a five, six, seven or eight is higher than the probability of other 7 numbers all together? We can tell things like that by adding the probabilities together, if the events we're talking about are completely independent, as they are in this case. So we can even write:
Pr(getting a 5, 7 or 8) = Pr(getting a 5) + Pr(getting a 7) + Pr(getting an 8)
But we can substitute these fractions and add them up easily to get:
Pr(getting a 5, 7, or 8) = (4+6+5)/36 = 15/36.
So now we know there is a 15 in 36 chance of having to pay rent to
the holder of the orange Monopoly! That simplifies to 5/12ths,
only a little less than
50%. Yikes!
Since we know the probability of all possible events must equal one, we can actually use that to compute the probability:
Pr(getting a 2,3,4,6,9,10,11 or 12) = 1 - (15/36) = (36/36 - 15/36) = 21/36.
This way we didn't have to add up all those individual probabilities, we just subtracted our fraction from 1.
Mathematically we could say: Pr(some events) = 1 - Pr(all the other possible events)
You might want to take the time now to get two dice and roll them 3*36 = 108 times, recording the sum of them each time. The number you get for each sum won't be exactly 3 times the probabilities that we have computed, but it should be pretty close! You should get around 18 sevens. This is fun to do with friends; you can have each person make 36 rolls and then add together all your results for each sum that is possible.
The fact that if you make a lot of rolls it is likely to get results close the probability of an outcome time the number of rolls is called the Law of Large Numbers. It's what ties the math of probability to reality, and what let's you do math to make a good decision about what roll you might get in a game, or anything else for which you can compute or estimate a probability.
Here's a homework exercise for each of you: compute the probabilities of every possible sum of throwing two dice, but two different dice: one that comes up 1,2,3 or 4 only, and the other that comes up 1,2,3,4,5,6,7 or 8. Note that we these dice (which you can get at a gaming store, if you want) the lowest and highest possible roll is the same but the number of possible rolls is only 4 * 8 = 32.
***| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
=====================================
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11
4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 12
Pr(getting a 2) = 1 / 32
Pr(getting a 3) = 2 / 32
Pr(getting a 4) = 3 / 32
Pr(getting a 5) = 4 / 32
Pr(getting a 6) = 4 / 32
Pr(getting a 7) = 4 / 32
Pr(getting a 8) = 4 / 32
Pr(getting a 9) = 4 / 32
Pr(getting a 10)= 3 / 32
Pr(getting a 11)= 2 / 32
Pr(getting a 12)= 1 / 32
(You should check our work by making sure that all these probabilites sum to 32/32 = 1).
So this is interesting: Although the possible scores of the throwing two dice are the same, the probabilites are a little different. Since 1/32 is a little more than 1/36, you are actually more likely to get a two or a twelve this way. And since 4 /32 = 1/8 is less than 6 / 36 = 1/6, you are less likely to get a seven with these dice. You might have been able to guess that, but by doing the math you know for sure, and you even know by how much, if you know how to subtract factions well.
So let's review what we've learned:
We know what the word probability means.
We know probabilities are always numbers between 0 and 1.
We know a basic approach to computing some common kinds of probabilites.
We know that the probabilities of all possible outcomes should add up to exactly 1.
We know the probability of a result is equal to one minus the
probability of all other results.
Watch out for the orange when you're on St. Charles!
--Robert L. Read, PhD 05:15, 27 Oct 2004 (UTC)