GMAT Exam: A Simple Guide To Arithmetic - Basic Probability

Basic Concepts
A random experiment is a chance process that gives a single result that cannot be determined beforehand.

For example, tossing a standard six-sided die and observing the up face is a random experiment.

Tip: A standard six-sided die is a balanced cube for which each of the six faces has one, two, three, four, five, or six dots on it.

An outcome is a single result from a random experiment.

When you toss a standard six-sided die, the six possible outcomes are 1, 2, 3, 4, 5, or 6, where “1” means “one dot on the up face,” “2” means “two dots on the up face,” and so forth.

An event is a collection of one or more outcomes.

For instance, when you toss a standard die, the event E that the die shows a number less than 3 consists of the outcomes 1 and 2. Tip: Uppercase letters represent events.

The probability of an event is the likelihood the event will occur.

Outcomes are equally likely if each outcome is as likely to occur as any other outcome. When you toss a standard die, 1, 2, 3, 4, 5, and 6 are equally likely outcomes.

An event is certain to occur if and only if the probability of the event is 1.

For example, when you toss a standard six-sided die, the probability a whole number of dots will show on the up face is 1.

An event is impossible if and only if the probability of the event is 0. For example, the probability the die will show seven dots on the up face is 0. The probability of any event is a number between 0 and 1, inclusive. Thus, the lowest probability you can have is 0, and the highest probability you can have is 1. All other probabilities fall between 0 and 1.

The closer the probability of an event is to 1, the more likely the event is to occur; and the closer the probability of an event is to zero, the less likely the event is to occur.

Compound Events
A compound event is a combination of two or more events.

The event A ∩ B (read as “A intersection B”) is the event consisting of all outcomes that A and B have in common.

The event A ∪ B (read as “A union B”) is the event consisting of all outcomes that are in A only, B only, or in their intersection.

Specifically, if an outcome is in A ∪ B, it is in at least one of the events A or B. The event (read as “complement of E”) is the set of outcomes that are not in E. It is the event that E does not occur. Tip: is also known as the opposite of the event E.

For example, suppose the possible outcomes for a random experiment are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Let event A consist of outcomes 2, 3, 5, and 7, and event B consist of outcomes 3, 5, 7, 9.

Then event A ∩ B consists of outcomes 3, 5, and 7; event A ∪ B consists of outcomes 2, 3, 5, 7, and 9; and event consists of outcomes 1, 4, 6, 8, 9, 10.

Mutually Exclusive Events
Two events are mutually exclusive if they have no outcomes in common, meaning the two events are disjoint. They cannot occur at the same time. The occurrence of one prevents the occurrence of the other.

For instance, when you toss a standard die, the event M that the die shows an even number (2, 4, or 6) and the event N that the die shows an odd number (1, 3, or 5) are mutually exclusive.

Probability Formula
If all outcomes are equally likely, the probability of an event E, denoted P(E), is


For example, for the die-tossing experiment, and the event E that the die shows a number less than 3,


The denominator for the probability of an event is always larger than or equal to the numerator.

Check for this requirement when you calculate probabilities.

Probability of the Complement of an Event
The probability of the complement of an event E is ; and, conversely, .

For example, if , then .

If the probability of an event is difficult to compute, try to find the probability of the opposite of the event, then subtract from 1.

The Addition Rule
For two events A and B, the event A ∪ B is the event that A occurs or B occurs or that both occur simultaneously on one trial of an experiment.

The Addition Rule states that P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

Tip: Keep in mind that this rule applies to one trial of an experiment.

For example, given the following probabilities for tomorrow’s weather: P(rain) = 0.7, P(temperature below 32°F) = 0.3, and P(rain and temperature below 32°F) = 0.15, then P(rain or temperature below 32°F) = 0.7 + 0.3 – 0.15 = 0.85.

In many situations, you must calculate the probabilities used in the addition rule.

For example, you toss a standard six-sided die one time. Let event A be the outcome is even, and event B be the outcome is greater than 4.

Then:


When you can determine the possible outcomes, an efficient way to find P(A ∪ B) is to sum the number of outcomes favorable to A and the number of outcomes favorable to B, being sure to add in such a way that no outcome is counted twice, and then divide by the total number of possible outcomes.

Applying this strategy to the previous example, there are 3 outcomes that are even (namely, 2, 4, and 6) and 1 outcome greater than 4 that is not even (namely, 5). So, there are 3 + 1 = 4 distinct outcomes favorable to the event “outcome is even or greater than 4,” Thus, P(outcome is even or greater than 4) .

When two events A and B are mutually exclusive, P(A ∩ B) = 0. So, for mutually exclusive events, P(A ∪ B) = P(A) + P(B).

If you toss a standard six-sided die:


When you want to find the chance that at least one of two events happens, use the addition rule.

Conditional Probability and Independent Events
P(B | A) (read as “Probability B given A”) is the conditional probability of event B, given that event A has already occurred. For P(B | A), you must compute the probability of event B by taking into account that the event A has already occurred.

For example, suppose that you randomly draw two marbles, one after the other, from a box containing 10 red marbles and 5 blue marbles.

Then the probability of drawing a blue marble on the second draw given that a red marble was drawn without replacement on the first draw is (because after the red marble is drawn without replacement, there are 9 red marbles and 5 blue marbles in the box.)

Tip: Without replacement means an object is NOT put back before the next object is selected.

On the other hand, the probability of drawing a blue marble on the second draw given that a red marble was drawn with replacement on the first draw is (because after the red marble is drawn and then replaced, there are 10 red marbles and 5 blue marbles in the box.)

Tip: With replacement means an object is put back before the next object is selected.

Two events A and B are independent if P(B) = P(B|A) and P(A) = P(A|B).

In other words, A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.

For instance, if you randomly draw two marbles, one after the other, from a box containing 10 red marbles and 5 blue marbles, the event of drawing a red marble with replacement on the first draw and the event of drawing a blue marble on the second draw are independent events.

If events A and B are not independent, they are dependent.

For example, if you randomly draw two marbles, one after the other, from a box containing 10 red marbles and 5 blue marbles, the event of drawing a red marble without replacement on the first draw and the event of drawing a blue marble on the second draw are dependent.

When you draw at random with replacement, the draws are independent.

When you draw at random without replacement, the draws are dependent.

The Multiplication Rule

The multiplication rule says that for two events A and B, the probability that event A occurs on the first trial and event B occurs on the second trial of an experiment is P(A)P(B | A).

This rule is used to find the probability of two events that occur in sequence.

Tip: Keep in mind that this rule applies to two trials of an experiment.

An efficient way to find the probability that event A occurs on the first trial and event B occurs on the second trial is to multiply the probability of event A times the probability of event B, where you have determined the probability of B by taking into account that the event A has already occurred.

For example, suppose you draw two marbles, one after the other, without looking, from a box containing 10 red marbles and 5 blue marbles.

Tip: A draw “without looking” is a random selection.

The probability of drawing a red marble on the first draw without replacement and a blue marble on the second draw is


When two events A and B are independent, the probability that event A occurs on the first trial and event B occurs on the second trial of an experiment is P(A) P(B).

For example, the probability of getting two heads on two flips of a coin is (because each flip of the coin is independent of the other).

When you want to find the probability that both of two events will happen, use the multiplication rule.