Sample GMAT Question: Probability

Written by Kelly Granson. Posted in GMAT Sample Questions

marblesProbability may not be the most frequently tested topic on the GMAT, but you are guaranteed to see at least one or two GMAT probability questions on the test. You can learn all the theory, but unless you practice with actual questions, the GMAT exam is going to be rather difficult.

The GMAT does not really test how much you know; what it does test is how well you can apply what you know. Practice question are therefore a vital part of any GMAT preparation, so let's practice.

Here is a typical GMAT probability question. Try to complete this sample GMAT question on your own before looking at a detailed breakdown below.

In a bag there are 3 white marbles and 2 black marbles. What is the probability of drawing at least one white marble when drawing two consecutive marbles randomly?

A) 13.09

B) 13.091

C) 13.092

D) 13.093

E) 13.094

Solution:

This question deals with the probability of drawing an item from a set. The complication here is that we need to draw two items from the bag, and the color of each of those items matters.

There are two ways we can solve this question.

Calculating the probability for all the combination in which a white marble is drawn

Calculating the probability that a white is not drawn.

We will look at each of these solutions in turn. Let's start with the solution most would attempt, calculating the probability for all the combination in which a white marble is drawn.

Method 1:

Think about all the ways, that the marbles can be drawn.

We could have:

1. Drawing white, white

2. Drawing white, black

3. Drawing black, white

4. Drawing black, black

One, two and three reflect the possibilities that the question asks us to find. Possibility number four does not include a white marble and therefore does not count towards our result. Therefore, it seems logical to work out the probability of methods 1,2 and 3 to find our answer.

Let's start with possibility number one, in which the two marbles drawn are white.

The formula for probability is:

CodeCogsEqn copy_copy_copy

Rearranging this to give us the probability of picking a white first gives:

CodeCogsEqn (1) copy_copy

CodeCogsEqn (2) copy_copy

Now we know the probability of the first marble being white, we can calculate the second marble.

Remember, now there is one less marble in the bag and importantly the marble we removed was white. So now there are four marbles left in the bag and only two of them are white.

Using the same equation we would get:

CodeCogsEqn (4) copy_copy

CodeCogsEqn (5) copy_copy

We now know the probability of picking a white marble first and then picking a white marble second. Now we need to find the total probability.

All we need to decide is whether to add the probabilities or multiply them. Remember, the clue is often in the wording.

The two events must occur together, thus an 'and' was used. Whenever events must occur together, we simply multiply the individual probabilities.

CodeCogsEqn (6) copy

Using the same method we can calculate the probability of drawing 'black, white' and 'white, black' respectively.

CodeCogsEqn (7) copy

CodeCogsEqn (8) copy

That gives us the probability of picking either

1. White, white

2. White, black

3. Or black, white

Either of these combinations fulfils the requirement of the question. However we still have three probabilities and we need to decide whether to add them probabilities or to multiply them to find the correct answer. These events do not have to occur together; any of them occurring is enough to satisfy our questions, so in cases such as this, we add the probabilities together.

The final answer is therefore:

Prob of picking at least one white = (Prob. of W.W.)+ (Prob. of B.W.) + (Prob. of W.B.)

Or numerically:

CodeCogsEqn (9) copy

Method 2:

Although method 1 is straightforward, there is still a quicker way to arrive at the same answer.

Method 2 words in reverse and relies on one simple understanding; that the sum of probabilities equals 1. In other words:

P (will occur) = 1 – P (will not occur)

If we look back at the four possibilities we formulated, we have:

1. Drawing white, white

2. Drawing white, black

3. Drawing black, white

4. Drawing black, black

Possibility number four was the only one that did not satisfy the condition of drawing of at least one white marble, and that is why we ignored it in method 1. In method two, however, we only use this possibility and ignore all the others.

P (at least one White)= 1-P(no White)

Drawing black, black is the only possibility that does not produce at least one white marble. The probability of drawing black, black is:

CodeCogsEqn (10) copy

Replacing that in the equation gives us:

CodeCogsEqn (11) copy

As you can see, both methods bring us to the same result, but the second one is more efficient. Method 2 is naturally slightly quicker, but requires a better understanding of probability. It is wise to learn and practice method two, as it will save you time in the GMAT exam, and certain higher-level question may rely upon your understanding of this concept.

Remember:

P(will) = 1 – P (will not)

Similarly:

P(will not) = 1 – P (will)

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