Probability

The SBC GMAT Files

Probability
Staff, Test Prep New York/Test Prep San Francisco

Say what you will about statistics, but an understanding of probability is one of the most useful tools that you can have at your disposal, both on the GMAT and in your upcoming MBA program. You can’t use them to predict the future, but probabilities can still give you a huge amount of useful information and help you maximize the score of the quantitative section of your GMAT.

So What is Probability, Anyway?

In simple terms, a probability tells you how likely a particular event is to occur or not to occur, and is measured either as a percentage between zero and 100, or as a decimal between zero and one. Multiple probabilities for a single event must always sum to 100 or one, depending on which format you are using. Probabilities do not guarantee something will or will not happen, but instead explain how likely something is to occur.

To illustrate how probabilities work, and to learn a few basic operations, let’s consider the following statement:

“According to ESPN, the New Orleans Saints have a 75% chance of winning their next football game against the Atlanta Falcons.”

It looks like the Saints have a pretty good chance of coming out on top, but we can take this sentence a lot deeper. To start, we can sum this sentence up in mathematic terms in two possible ways:

P(Saints Win) = 75%
P(Saints Win) = .75

Believe it or not, those each say the same thing as our example sentence above. The “P” simply stands for “probability,” and whatever is in the parenthesis tells us what we are getting the probability of. 75% and .75 (which are actually 75 / 100 or 0.75 / 1 respectively) represent the same probability; we simply have two different ways to present the same information. You will likely use decimal form for probabilities in your b-school classes, so we’ll proceed with those from now on.

Although the example sentence didn’t tell us anything about the Falcons themselves, we can still calculate their probability of winning based on the information we have. Remember that the probabilities for a given event must always sum to one, so we can do a little simple algebra to learn what the Falcons’ probability of winning is:

P(Saints Win) + P(Falcons Win) = 1
.75 + P(Falcons Win) = 1
(-.75)                (-.75)
P(Falcons Win) = .25

In other words, the Falcons have a 25%, or .25, chance of winning the game. We now have a complete picture of each team’s chance of winning the game, even though ESPN only talked about the Saints.

In some cases you can use probabilities as a relative measure, as well; for example, because the Saints’ chance of winning is .75 and the Falcons’ is .25, we can say the Saints are three times as likely to win as the Falcons (because .25 x 3 = .75). However, even with these operations complete, we still don’t have the full picture.

Although the Saints have a much higher chance of winning than the Falcons, these probabilities do not guarantee that the Saints will win. Remember, the Falcons have a .25 chance of victory! The only thing that these or any probabilities tell us for sure is that the Saints will probably win the upcoming game, and that they are more likely to win than the Falcons are. That’s it. The probabilities do not guarantee any specific outcome, so the only way to know the outcome for sure is to watch the game and find out!

Different Types of Events

Now that we understand the basics, we can dive a little deeper and talk about probabilities in a sequence of events, where the same action occurs multiple times. There are two types of events we can deal with, independent and dependent.

The first type are Independent Events. There are events that occur in a series but do not effect each others’ outcomes. If we return to our football example from before, the Saints’ entire season of football games is a series of independent events because whether the Saints win or lose one game has no bearing on whether they win or lose the next game. In addition, if you flip a coin multiple times, each individual coin flip is an independent event, because none of the flips can influence the outcome of any other flip.

Meanwhile, Dependent Events are events that occur in a series in which the outcome of each event is influenced by the outcome of the previous events. For example, if you are dealing cards, the probability that you will get an Ace (of which there are four in the deck) changes each time you deal a card. It starts as 4 / 52, then moves to 4 / 51 after you deal the first card, 4 /50 for the second card, 4 / 49 for the third, and so forth. If your fourth card is an Ace, the probability of getting another Ace would then be 3 / 48. Each time you deal a card, you change the probability of your next deal.

To illustrate how we work with each, consider the following example:

Bill has a bag of eight marbles: one red, four blue, two green, and one yellow. When he is looking at his collection, he takes one marble out of the bag, looks at it, and puts it back before picking out another marble. What is the probability that Bill will pull out the red marble? What is the probability he will pull out the red marble, then the blue marble in that order?

To start out, let’s get our basic probabilities down. The question only asks us about red and blue marbles, so we can focus in on those and calculate their probabilities to start:

P(Red) = One Red in Eight Marbles = (1/8) = 0.125
P(Blue) = Four Blues in Eight Marbles = (4/8) = .50

Because Bill puts each marble back in the bag before pulling out another, the total number of marbles in the bag stays the same and none of his selections will affect the probability of the next. That means that we’re dealing with independent events, so the answer to our first question, the probability of drawing a red marble, is simply .125, as we calculated above. It doesn’t matter where Bill starts, because he puts the marbles back in the bag after each draw, he always has a .125 chance of drawing a red marble.

For the second part of the question, order is important, so things will get a little more complicated. We need to know the probability that Bill will draw a red marble, then put it back in the bag, then immediately draw a blue marble right after. Calculating the probability of two independent events occurring together is easy; all you need to do is multiply the two individual probabilities together. In math speak:

P(Red) x P(Blue) = P(Red and Blue)
.125 x .50 = P(Red and Blue)
.0625 = P(Red and Blue)

In other words, there is a .0625 chance that Bill will pull a red marble, then a blue marble, or in plain English, Bill will pull a blue after a red 6.25% of the time. It doesn’t matter how many events occur in your sequence; if they are independent you need only multiply their probabilities together to get your answer.

Now let’s see what would happen if Bill decides to switch things up on us:

Bill decides to pull out five marbles at once. What is the probability that the order will be red, blue, yellow, blue, green?

In this case, the marbles now come out and stay out, meaning that each time Bill pulls a marble out, he changes the total number in the bag and affects the probability of the next pull. That means we are now dealing with dependent events, so we have to adjust our calculation somewhat. In general terms, because order is important, we still have to multiply our probabilities together, but this time they change. We have to recalculate our probabilities given the fact that the number of marbles is decreasing, which we can do as follows. First, let’s set up the basic equation and plug in some numbers:

P(Red) x P(Blue) x P(Yellow) x P(Blue) x P(Green) =P(Order)
(1/8) x (4/7) x (1/6) x (3/5) x (2/4) = P(Order)

So where are these numbers coming from? Let’s look at the denominators first. Notice how they start at 8, then move to seven, then six, then five, then four? That is the total number of marbles in the bag, which gets smaller each time Bill pulls a marble out. The numerators are simply telling us how many of each color are in the bag, but notice that the first blue is 4 and the second blue is 3: because we are not replacing marbles, there will only be 3 blue marbles left in the bag after we draw the first blue marble. The fact we are not putting marbles back is changing both sides of our fraction.

Now that we see where the numbers are coming from, let’s complete the calculation:

P(Red) x P(Blue) x P(Yellow) x P(Blue) x P(Green) =P(Order)
(1/8) x (4/7) x (1/6) x (3/5) x (2/4) = P(Order)
(.125) x (.571) x (.167) x (.60) x (.50) = P(Order)
.00358 = P(Order)

In other words, given that Bill does not put marbles back in the bag, he has a probability of .00358, or 0.358% chance, of pulling marbles out in that exact order. The low probability means getting that exact order is unlikely, but remember, it’s not impossible!

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