GMAT Probability: The Word "Or"

March 13, 2008

One of the trickier concepts in GMAT Probability is the meaning of the word "or." In fact, "or" represents two concepts, and the first challenge is learning to tell them apart.

Simultaneous and Non-Simultaneous Probabilities

What I'm talking about are questions that are looking for the probability of either A or B happening. The subject could be flipping a coin twice and getting tails the first time or the second time; choosing a marble from a bag and getting a blue marble or a red marble; selecting a winner from a pool of contestants and awarding the prize to either Franka or Gene.

Two of those three examples are nearly identical; the other one is different. Can you tell which?

The key idea here is the possibility that both A and B will occur. When you flip a coin twice, you could get tails both the first and second time. If you choose a marble from a bag, though, it's impossible (nonsensical, even) to pick both a blue and a red. Similarly, if you select one person, Franka and Gene can't both win.

Simultaneous: The Easy One

If you've learned some probability, you've probably heard the soundbite version of how to deal with "or:" add. Take the following example, in which we're looking for the probability of one or the other event occuring, and there is no possibility that both will occur:

A bag consists of 20 marbles, of which 5 are blue, 9 are red, and the remainder are white. If Lisa is to select a marble from the bag at random, what is the probability that the marble will be red or white?

One way to approach this is to calculate each individual probability. There are 20 marbles, including 5 blues, 9 reds, and 6 (subtract 5 and 9 from 20) whites. The probability of selecting a red is 9/20, and the probability of selecting a white is 6/20, or 3/10.

Since the two events cannot possibly both occur, the probability of one or the other occuring is their sum: 9/20 + 6/20 = 15/20, or 3/4.

While that isn't terribly time-consuming or demanding in any way, recognize that it's possible to find the answer by solving for the opposite. Instead of finding the probability of getting a red or a white, find the probability of not getting a red or a white.

The only way that happens is if a blue is selected, and the probability of a blue being chosen is 5/20, or 1/4. The probability of something happening is 1 minus the probability that is doesn't happen, so the probability we're looking for is 1 - 1/4 = 3/4.

Non-Simultaneous: Things Get Complicated

Let's look at another sample question, again with the marbles:

Mario and Nina each have a bag of marbles, each of which contains 4 blue marbles, 10 red marbles, and 6 white marbles. If Mario and Nina each select one marble from their respective bags, what is the probability that either Mario or Nina select a red marble?

It doesn't take long to realize that addition doesn't work here. There's a 10/20 (one-half) chance of selecting a red marble out of either bag, and if you add the probability of Mario selecting a red marble (1/2) to the probability of Nina selecting a red marble (1/2), the result is 1. Surely it isn't guaranteed that one or the other of them selects a red marble!

Because the two events are non-simultaneous--that is, because it's possible that both can occur--what we've done is double-counted the probability that both occur. A similar scenario would occur if we were told that there are 20 people in chess club and 30 people in math club, and we assumed that there were 50 different people in either chess or math club. That's true if no student is in both clubs, but if it's occuring on a GMAT question, there is likely to be some overlap.

Overlapping Probabilities

For both Mario and Nina, there is a 1/2 chance they select a red marble, and a 1/2 chance they do not. There are four possible results:

• Mario and Nina both select red marbles.
• Mario selects a red marble, and Nina does not.
• Nina selects a red marble, and Mario does not.
• Neither Mario nor Nina selects a red marble.

That offers us two possible solutions. First, we could find the probability of each of the first three scenarios, and add them together. (It's impossible for more than one of the first three scenarios to occur.) Second, we could find the probability of the final scenario, and subtract it from 1.

Which one do you think I would recommend?

You guessed it: let's stick with one probability. The word "and" signals multiplication, so find the individual probabilities of each person not selecting a red marble. We already know that each is 1/2, so the probability that neither selects a red marble is (1/2)(1/2) = 1/4.

The answer, then, is 1 - 1/4 = 3/4.

Summary

Any time you see the word "or" in a probability question, you'll need to find multiple probabilities. If there is no possibility that both will occur--that is, they are simultaneous--you can simply add them and find the answer. If it's possible that both will occur, consider all of the possible ways in which the probabilities combine, or just find the probability of the opposite, which usually requires much less work.

There are a number of probability questions, including a few that use "or," in The Official Guide to the GMAT. There are many probability items in my "Extreme Challenge" practice set, as well as an entire chapter on probability in my Total GMAT Math.

About the author: Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular Total GMAT Math and Total GMAT Verbal, as well as 1,800 practice GMAT math questions.

• Older: Understanding the Difference Between Combinations and Permutations
• Newer: Ten Important GMAT Math Topics You Must Master
• General GMAT Math: more articles
• All articles at GMAT HACKS