GMAT

Probability

Probability questions are becoming increasingly common. Probability questions tend to be bundled among the difficult questions, so high scorers will commonly encounter them. If you are a low scorer and are pressed for time, consider skipping most of the material past "Simple Probability." This is a computer-adaptive test, and low scorers aren't likely to encounter the most difficult probability question types.

A. Simple Probability
B. Probability of Multiple Events
C. Independent and Dependent Events
D. Mutually Exclusive Events
E. Conditional Probabilities
F. Combinations

A. Simple Probability

In general, the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

Example 1

What is the probability that a card drawn at random from a deck of cards will be an ace?

Solution
In this case there are four favorable outcomes:
(1) the ace of spades
(2) the ace of hearts
(3) the ace of diamonds
(4) the ace of clubs.

Since each of the 52 cards in the deck represents a possible outcome,
there are 52 possible outcomes. Therefore, the probability is 4/52 or 1/13.

The same principle can be applied to the problem of determining the probability of obtaining different totals from a pair of dice.

Example 2

What is the probability that when a pair of six-sided dice are thrown, the sum of the numbers equals 5?

Solution
There are 36 possible outcomes when a pair of dice is thrown. Consider that if one of the dice rolled is a 1, there are six possibilities for the other die. If one of the dice rolled a 2, the same is still true. And the same is true if one of the dice is a 3,4,5, or 6. If this is still confusing, look at the following (abbreviated) list of outcomes: [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6); (2,1),(2,2),(2,3)… (3,1),(3,2),3,3)… (4,1)…(5,1)…(6,1)….

The total number of outcomes is 6 × 6 = 36. Since four of the outcomes have a total of 5 [(1,4),(4,1),(2,3),(3,2)], the probability of the two dice adding up to 5 is 4/36 = 1/9.

Example 3

What is the probability that when a pair of six-sided dice are thrown, the sum of the number equals 12?

Solution
We already know the total number of possible outcomes is 36, and since there is only one outcome that sums to 12, (6,6--you need to roll double sixes), the probability is simply 1/36.

NOTE: The material from here on through the end of the section is dense and intended only for medium to high scorers. Because this is a CAT (computer-adaptive test), it is relatively unlikely that lower scorers will encounter these questions, and, if they are short of time, they are better off putting their time into other sections.

Hard Question:

1. A book contains 732 pages numbered 1, 2, ..., 732. If a student randomly opens the book, what is the probability that the page number contains digit 1?

We have to compute how many numbers between 1 and 732 inclusive contain digit 1. In the first 99 there are 19 such numbers: 10 numbers where 1 is a units digit (1, 11, 21, ...), 10 numbers where 1 is a tens digit (10, 11, 12, ...), minus 1 number (11) which is present in both sets.

It is clear that there are also 19 numbers containing digit 1 in each of the following number ranges: 201-299, 301-399, ..., 601-699. So far, we have counted 19×6 = 114 pages containing digit 1. To this number we have to add 100 pages from range 100-199 and 13 pages from range 701-732. In all, there are 114 + 100 + 13 = 227 pages in the book which contain digit 1. Thus, the required probability = _.