Probability: A Formulaic Approach for the GRE

When a question asks us to determine the probability of a certain event occurring, we need to ask ourselves two questions:
  1. How many possible outcomes are there?
  2. How many of those possible outcomes will fulfill the requirements given?
The answers to the above questions will allow us to determine probability using the formula:
Let’s look at a GRE question that requires us to apply this formula:
What is the probability of tossing a 1 or a 3 on a fair 6-sided die?
(Notice that the GRE does not leave you wondering about whether the die is fair or not—they remove that doubt right away. Don’t spend time entertaining “what ifs” that have already been resolved.)
If we toss a 6-sided die, how many possible outcomes are there?
Well, we could get a 1, 2, 3, 4, 5 or 6. There are 6 possible outcomes.
How many of those outcomes would fulfill the requirements given (i.e., be a 1 or a 3)? Only 2—a 1 or a 3.
Plugging those numbers into the probability formula, we get:
There is a 1/3 probability of tossing a 1 or a 3. That means we would expect a 1 or a 3 to show up in one out of every three tosses of the die.
That was a pretty easy example. Let’s look at a question that is a bit more complex:
A bag contains 3 red marbles and 5 blue marbles. What is the probability of randomly drawing a red marble, replacing the red marble back in the bag, then randomly drawing a blue marble?
In this problem, we have to calculate two probabilities—the probability of drawing a red marble and the probability of drawing a blue marble.
Since there are 3 red marbles and 5 blue marbles, we know there are 8 marbles in the bag. Therefore, the number of possible outcomes is 8. That will be true for both probabilities because we are replacing the red marble back in the bag before drawing a second marble. Also, because we are returning the red marble to the bag, these probabilities are independent—the color of the second marble drawn will not be determined in any way by the color of the first marble drawn.
Using the probability formula again, we find the following probabilities:
In order to find the probability of drawing a red marble, then drawing a blue marble, we have to multiply the two independent probabilities. Doing so, we find that the probability of red, then blue is:
Since 15 and 64 do not have any common factors, we cannot simplify the fraction any further. Thus, we would expect to draw a red marble, then a blue one, 15 out of every 64 selections of two marbles.
Of course, GRE probability questions can be much more complex than these examples. Be on the lookout for my next probability post, which will focus on the probability of flipping a coin or tossing a die.