# Probability - GMAT Math Study Guide

## Definitions

• Event - An outcome to a random occurrence.
For example, the following are events: drawing a black card from a 52-card deck, a coin landing on heads, rolling an even number on a 6-sided die.
• Probability, P(A) - The likelihood of event A occurring.
For example, the probability of a coin landing on heads is 0.5 since there is a 50% chance (i.e., probability) of the coin turning up heads when flipped.
• Intersection of Events, P(A∩B) - When two events are fulfilled simultaneously.
For example, the intersection of the events "rolling an even number" and "rolling a number less than three" is rolling a 2 since rolling a 2 fulfills both events (i.e., 2 is both even and less than 3).
• Union of Events, P(A∪B) - When either of two events is fulfilled.
For example, the union of the events "rolling an even number" and "rolling a number less than three" is rolling a 2, 4, 6 (an even number) or rolling a 1, 2 (a number less than three).
• Dependent Event - An event whose probability of occurring is influenced by (i.e., dependent on) whether another event occurs.
For example, the probability of drawing a red card from a normal 52-card deck after you draw another card from the 52-card deck without replacing this other card is a dependent event. The probability of the second card you draw being red depends on what card was drawn the first time. In particular, the probability of the second card being red depends on whether you drew a red card the first time (in which case one less red would be in the deck). A dependent event is the opposite of an independent event.
• Independent Events - Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other.
For example, the probability of flipping a coin twice and the coin landing on heads the second time is not affected by (i.e., is independent of) whether the first coin flip turned up heads or tails. An independent event is the opposite of a dependent event.
• Conditional Probability, P(A|B) - The likelihood of event A occurring given that event B already occurred.
For example, the probability of drawing a red card from a 52 card deck is not the same as the probability of drawing a red card from a 52 card deck given that you have already drawn three red cards and did not put them back into the deck. This later probability is an example of conditional probability.
• Mutually Exclusive Events - Two events are mutually exclusive if they cannot occur together.
For example, the events "rolling an even number" and "rolling an odd number" are mutually exclusive since by definition rolling an even number means you cannot roll an odd number. By contrast, the events "rolling an odd number" and "rolling a number two or less" are not mutually exclusive since you could roll the number one, which is both odd and a number two or less.
• Complement of an Event, P(A') or P(AC) - The event that is composed of all the outcomes that are not in another event.
For example, the complement of flipping a coin and it landing on heads is flipping a coin and it landing on tails. The complement of rolling an even number is rolling an odd number. The complement of drawing a heart is drawing either a spade, club, or diamond. The complement of the temperature being 30 degrees is the temperature not being 30 degrees.

### Graphical Representation

The following table elucidates the relationship between the different types of events.

 Mutually Exclusive Yes No Independent Yes Impossible Possible No Possible Possible

The following chart shows the relationship between the intersection of two events, the union of two events, and the complement of an event. In each case, the probability in question is represented by the area in gray. For example, the gray in the middle of the far left section of the graph represents the intersection of events A & B. ## Basic Probability

The probability of an event occurring is the likelihood of it happening expressed in mathematical terms. The likelihood of an event occurring is the number of ways that particular event can occur divided by the number of ways any possible outcome can occur. Said differently, the probability of an event occurring is the number of ways the specific event outcome can occur divided by the number of ways any possible event can occur. Said one other way, the probability of an event is the number of outcomes that fulfill that event divided by the number of total possible outcomes. Consider the following example:

What is the probability of rolling an even number on a die?

Outcomes Meeting the Criteria: roll a 2, 4, or 6
Possible Outcomes: roll a 1, 2, 3, 4, 5, or 6
Probability of rolling an even number: 3/6 = .5

Consider another example:

If a bucket contains all the factors of 20 and each unique factor appears only once in the bucket, what is the probability of drawing a prime number?
Possible Outcomes: 2, 4, 5, 10, 20
Outcomes Meeting the Criteria: 2, 5
Probability of drawing a prime number: 2/5 = 2 outcomes meeting criteria / 5 possible outcomes

### 0 < P(A) < 1

While calculating the probability of certain events can be complex, there is one rule that always applies:

0 < P(A) < 1

Regardless of the circumstances, the probability of an event occurring can never be less than zero or greater than one. The rationale for why the probability of A occurring can never be greater than one is that this would require the numerator being larger than the denominator in the above formula for P(A). Since the denominator includes the numerator (i.e., the total number of ways any outcome can occur includes the number of ways outcome A can occur), it is impossible for the numerator to be larger than the denominator. Similarly, a value for P(A) less than zero would require a negative number of ways an outcome could occur and this is logically impossible.

## Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur simultaneously. In other words, if event A occurs, then event B cannot occur. There is no overlap between events A and B if these events are mutually exclusive. P(A∩B) = 0 for all mutually exclusive events A and B

Examples of mutually exclusive events:

• Drawing a red card and drawing a black card
• Rolling an even number and rolling a 3

## Conditional Probability

The concept of conditional probability pertains to the probability of a certain event occurring given that another event already occurred. The concept of conditional probability is important since the occurrence of one event can change the probability of another event.

The conditional probability of A given B, denoted P(A|B): Consider the following example:

If the probability of choosing a red marble from a bag of 10 marbles is 6/10 and the probability of choosing a red marble and then a green marble is 3/10, what is the probability of choosing a green marble if you already selected a red marble?

Let A = the probability of selecting a green marble
Let B = the probability of selecting a red marble on your first selection

P(A∩B) = 3/10 = .3 as given in the problem
P(B) = 6/10 = .6
P(A|B) = .3/.6 = .5

## Independent vs. Dependent Events

Two events are independent if the occurrence of one event does not change the probability of the occurrence of the other event. For example, if the probability of event A = x before event B occurs and the probability of event A = x after event B occurs, the two are independent.

Two events are dependent if the occurrence of one event does change the probability of the occurrence of the other event. For example, if the probability of event A = x before event B occurs but the probability of event A = y after event B occurs, the two events are dependent.

For Independent Events: P(A|B) = P(A)
For Two Dependent Events: P(A|B) ≠ P(A)

The following are examples of dependent events:

• The probability of drawing a red card on your first pick and the probability of drawing a black card on your second pick from a normal deck of 52 cards when the first card is not replaced (i.e., when the first card drawn is not put back into the deck).
• With a bag of 10 marbles that consists of 3 yellow and 7 blue, the probability of drawing a yellow marble from the bag and a red marble from the bag without replacing the first marble chosen.

The following are examples of independent events:

• The probability of a coin landing on heads and the probability of a coin landing on tails. Whether the coin previously landed on tails makes no difference in calculating the probability that the next flip of the coin will land on heads since there is no relationship between the outcome of the flip of the coins.
• The probability of drawing a red card from a deck and the probability of drawing a blue marble from a bag of marbles. The two events have no relationship (i.e., one event occurring does not affect the probability of the other event occurring).

## Union of Events: Or The union of events includes not only the probability of A and B, but also the probability of only A and the probability of only B.

P(A∪B) = P(A) + P(B) - P(A∩B)

The last term is important since it removed items that appear in both A and B, thereby avoiding double counting.

Consider the following example, meant to clarify the importance of the - P(A∩B) term in the formula for P(A∪B).

When rolling a die, what is the probability of rolling a number greater than 1 or a number that is even?

The presence of the word "or" is a clue that the question is asking about the union of events.
Let Event A = rolling a number greater than 1
Let Event B = rolling a number that is even

P(A) = 5/6 since there are five possibilities and six total numbers that could be rolled
P(B) = 3/6 since there are three possibilities (i.e., 2, 4, 6) and six total numbers that could be rolled (i.e., 1, 2, 3, 4, 5, 6)
P(A∪B) ≠ 5/6 + 3/6 since: (1) this number is greater than one, which is impossible for a probability (2) this number double counts the even numbers greater than 1 (i.e., 2, 4, 6)

P(A∩B) = rolling a number greater than 1 and even = 3/6 since there are three possibilities (i.e., 2, 4, 6) and six total numbers that could be rolled (i.e., 1, 2, 3, 4, 5, 6)
P(A∪B) = 5/6 + 3/6 - 3/6 = 5/6

Consider another example:

When drawing a card from a normal 52-card deck, what is the probability of drawing a red card or an ace?

The presence of the word "or" is a clue that the question is asking about the union of events.
Let Event A = drawing an ace
Let Event B = drawing a red card

P(A) = 4/52 since there are four aces in each deck of 52 cards
P(B) = 1/2 = 26/52 since there are four suits and two of them are red (or 26 red cards in a deck of 52)
P(A∩B) = the probability of drawing a red ace = 2/52 since there are 2 red aces in a deck of 52 cards)
It is important to subtract P(A∩B), otherwise you would double count the red aces.

P(A∪B) = 4/52 + 26/52 - 2/52 = 28/52

A helpful way to memorize the particularities of the union of events is that the symbol, X, makes a U and the union brings together two probabilities.

## Intersection of Events: And The intersection of events includes only the probability that both events A and B occur simultaneously. If event A occurs alone or if event B occurs alone, this does not fall within the intersection of events A & B.

P(A∩B) = probability of the intersection of events A and B
P(A∩B) = P(A)P(B|A)
P(A∩B) = P(B)P(A|B)

If events A and B are mutually exclusive (i.e., if one event occurs then the other cannot occur), the formula can be simplified:

P(A∩B) = probability of the intersection of events A and B
P(A∩B) = P(A)P(B)

Consider a basic example:

What is the probability of drawing a red card from a normal 52 card deck and rolling an even number on a normal 6-side die?

Since drawing a red card and rolling an even number are entirely unrelated and have no bearing on each other, the two events are independent and you can use the simplified formula for the intersection of events.

Let Event A = drawing a red card
Let Event B = rolling an even number
P(A) = 1/2
P(B) = 1/2
P(A∩B) = P(A)P(B)
P(A∩B) = (1/2)(1/2) = 1/4

## Complement of an Event: Not The formula for the complement of an event, denoted P(AC):

P(A') or P(AC) = 1 - P(A)
Stated Differently: P(AC) + P(A) = 1

One common misconception that some students make is to assume that the complement of an event is simply its opposite. For example, it would be wrong to assume that the complement of a positive number is a negative number. Instead, the complement of a positive number is everything that is not a positive number (i.e., negative numbers and zero).

Consider a simple example:

If a coin is flipped 3 times, what is the probability that it lands on heads at least once?

While you could solve this problem by using the laws of independent events, this question can also illustrate the concept of the complement of an event. For most students, it is easier to find the probability of the complement of "a coin lands on heads at least once" than it is to find the probability that a coin "lands on heads at least once."
Let Event A = the coin lands on heads at least once
By Definition, Event AC = the coin never lands on heads

For the coin to never land on heads, a tail must be flipped every time. The probability of this is P(AC)
P(AC) = (1/2)(1/2)(1/2) = 1/8
The above formula for the complement of an event can now be used to find P(A):
P(AC) + P(A) = 1
1/8 + P(A) = 1
P(A) = 1 - (1/8) = 7/8

## Types of GMAT Problems

1. Calculating Probabilities of Independent Events

When trying to determine P(E∩F) it must first be established if E and F are independent or not. If E and F are independent the rule P(E∩F) = P(E)P(F) can be used. If the events are not independent then there is no way to simplify the problem.

Event E is defined to be rolling an even number on a 6-sided die and Event F is defined to be rolling a 1, 2 or 3. Calculate the probability of rolling a die such that events E and F occur simultaneously on a single roll of the die.
 A) 1/2 B) 1/6 C) 0 D) 5/6 E) 1
1. The question is asking for P(E∩F), which is the probability that events E and F are both satisfied by a single role of the die. In other words, we must find the probability that an even number from the set {1, 2, 3} is rolled.
2. There are two common ways to solve this problem: statistical formulas or intuition.
3. Method 1: Statistical Formulas
1. Determine if E and F are independent.
P(E|F) = the probability of event E occurring given that event F has already occurred
P(E|F) ≠ P(E) as P(E) is changed if it is given that F occurs.
Consequently, P(E∩F) ≠ P(E)P(F) since the events are not independent. As a result, P(E∩F) ≠ 1/4 = P(E)P(F)
2. We must use the more complex formula for finding the intersection of E and F:
P(E∩F) = P(F)P(E|F)
3. P(F) = the probability of event F occurring
The probability of rolling a 1, 2, or 3 is 3/6 = 1/2
4. P(E|F) = the probability of event E occurring given that event F has already occurred
If event F already occurred, we are limited to the numbers 1, 2, or 3. Within this range of 3 possible numbers, only one number--namely, 2--is an even number. Therefore, P(E|F) = 1/3
5. P(E∩F) = P(F)P(E|F) = (1/2) * (1/3) = 1/6
4. Method 2: Intuition
1. 2 is the only number of the six numbers on the die that can be rolled that will cause both events E and F to occur.
2. P(E∩F) = P(rolling a 2) = 1/6 since there are six numbers on the die and only one of these is even and either 1, 2, or 3.
2. Calculating Probabilities of Mutually Exclusive Events

P(E∪F) = P(E) + P(F) - P(E∩F) can be simplified to P(E∪F) = P(E) + P(F) when E and F are mutually exclusive events.

If E is defined to be drawing a black card out of a normal 52 card deck and F is defined to be drawing a heart, what is the probability of either E or F coming true?
 A) 1/2 B) 2/3 C) 1 D) 5/8 E) 3/4
1. The question is asking for P(E∪F), which is technically referred to as the union of events E and F (i.e., the probability of either event E or event F coming true).
2. The statistical formula for this type of problem is:
P(E∪F) = P(E) + P(F) - P(E∩F)
In words: the probability of events E or F = the probability of event E + the probability of event F - the probability of both events E and F occurring simultaneously
Note: It is necessary to subtract P(E∩F) in order to avoid double counting.
3. P(E) = 1/2 since all cards are either black or red and, as a result, half of the cards are black.
4. P(F) = 1/4 since there are four types of cards (i.e., hearts, diamonds, clubs, or spades).
5. P(E∩F) = 0 since it is impossible to pick a card that is both black and a heart because of the fact that hearts are red cards (not black cards).
6. P(E∪F) = P(E) + P(F) - P(E∩F)
P(E∪F) = (1/2) + (1/4) - 0 = 3/4
7. For students with a more advanced understanding of statistics, you will notice that E and F are mutually exclusive. As a result, a simpler statistical formula can be used:
E and F Mutually Exclusive
1. Determine if E and F are mutually exclusive.
P(E∩F) = 0. It is impossible to draw a card that is both black and a heart. Thus, the two events are mutually exclusive.
2. P(E∪F) = P(E) + P(F)
P(E) = 1/2 as half of a deck is composed of black cards while the other half is composed of red cards.
P(F) = 1/4 as one fourth of the deck is composed of hearts.
P(E∪F) = P(E) + P(F) = 1/2 + 1/4 = 2/4 + 1/4 = 3/4