In our last post, we talked about the idea of an experiment, outcome, and event. If you're not familiar with those concepts, it may be a good idea to look at that post. Here, we will talk about some of the basic features of probability. First, a definition:
Definition: The probability of an event is a number that measures the likelihood of the event occurring.
And because it is tedious to always write out things like "the probability that a fair coin lands leads is ½", we will adopt an abbreviation. We use letters to represent events:
E = A fair coin lands heads
And then, we just write:
P(E) = ½
which we read as:
The probability that “A fair coin comes up heads” is ½.
And in general, for any event E, we use P(E) to denote the probability that event E occurs. This shorthand will save us much space in the rest of the series.
Now, a probability measures the likelihood of an event. This brings us to:
5 Basic Facts About Probability
1. A probability of 0 means that an event is impossible.
So if you find that P(E) = 0, that means that E will not occur. As an example, when rolling a six-sided die, the event that we roll a 7 is impossible -- it does not occur in any of our outcomes. Thus, P(Roll a 7) = 0.
2. A probability of 1 means that an event is certain.
So, when rolling a six-sided die, the event that we roll some number is a certainty -- it occurs in all of our outcomes. Thus, P(Roll a number) = 1.
3. An event with a higher probability is more likely to occur.
So, if the probability that it snows is 20% while the probability that it rains is 80%, then it is more likely to rain than it is to snow. And, on the flip side, events with a lower probability are less likely to occur.
4. Probabilities are always between 0 and 1.
This makes sense, since if an event had a probability greater than 1, then it would be more likely to occur. But events with a probability of 1 are already certain to occur! How could any event be more likely than a certainty? Similarly, if an event had a probability less than 0, then it would be less likely to occur, but events with a probability of 0 are already impossible! How could an event be less likely than an impossibility?
This also gives us a helpful way to check our answers: if we get a probability greater than 1 or less than 0, we have made a mistake somewhere.
5. The probabilities of our different outcomes must sum to 1.
E.g. if we have 4 different outcomes, then
P(Outcome 1) + P(Outcome 2) + P(Outcome 3) + P(Outcome 4) = 1.
This is because, when we do an experiment, something is bound to happen. So the probabilities of our outcomes must sum to 1.
Now, for the GRE, there are three main types of probability problems:
- The probability of a single event occurring: P(A)
- The probability that two events both occur: P(A and B)
- The probability that one or another event occurs: P(A or B)
You are about to do an experiment with four possible outcomes: A, B, C, and D. The stated probabilities are as follows:
P(A) = .5
P(B) = .3
P(C) = .38
P(D) = .1
Is such an experiment possible? What if P(D) = -.18?
No, such an experiment is not possible since P(A) + P(B) + P(C) + P(D) = 1.28 which is not equal to 1. And changing P(D) would make P(A) + P(B) + P(C) + P(D) = 1, but then we would have P(D) = -.18, which is a negative number. Since probabilities cannot be negative, this experiment is again impossible.
Give an example of an experiment not discussed above, and give an example of an event with a probability of 0 for that experiment, and another event with a probability of 1 for that experiment.
There are many possible examples; here is one: investing in the stock market. It is a certainty that "The value of my portfolio will either increase, decrease, or stay the same." It is an impossibility that "The value of my portfolio will both increase and stay the same." It may do one, or the other, but to do both is impossible.
Translate P(A) + P(B) = ½ * P(C) into a natural language (like English, French, Chinese, etc.).
Again, there are many possible solutions. Not confident in my French, I'll stick with giving the English translation: "The probability of A plus the probability of B is equal to half of the probability of C."