Probability - GMAT PDF Download

Probability is a quantity that expresses the chance, or likelihood, of an event.
Think of probability as a fraction:

Probability represents the likelihood of an event occurring and can be visualized as a fraction. Let's consider a coin toss: when flipping a fair coin, the chance of getting heads is 1 out of 2 equally likely outcomes (heads or tails), expressed as a fraction, 1/2. In this context, the numerator (1) denotes the favorable outcome (heads), which is a subset of the total possible outcomes (2).

Probability values always range between 0 and 1. If there are n possible outcomes, the number of favorable outcomes must fall between 0 (indicating an impossible event) and n (representing a certain event). For example, an impossible event has a probability of 0, while a certain event has a probability of 1.

Expressing probability as a fraction, decimal, or percentage might be necessary. For instance, the probability of an event occurring can be shown as 3/4 (fraction), 0.75 (decimal), or 75% (percentage). Although questions might request probability in different forms, converting it to a fraction is often the initial step in calculations. This helps establish a foundational understanding before presenting the probability in various formats to suit specific requirements.

Calculate the Numerator and Denominator Separately

The relationship between the numerator and denominator in probabilities is intertwined, yet they're computed independently. In many cases, starting with the denominator calculation proves simpler.

Determining the number of outcomes for either the numerator or denominator involves two main methods:

• Utilizing combinatorics formulas: Employing established formulas tailored for counting outcomes in different scenarios. These formulas, such as combinations or permutations, provide structured ways to compute the total possibilities or favorable outcomes.
• Manual counting: Sometimes, it's necessary to manually count the outcomes. This involves going through the options one by one to ascertain the total number of possibilities or the specific outcomes that favor the event of interest.

Both methods serve as tools to determine the overall potential outcomes or the specific desired outcomes, allowing for the calculation of probabilities. While formulas offer structured approaches for complex scenarios, manual counting can be applied when the situation is more straightforward or when formulas aren't readily applicable.

Example: Two number cubes with faces numbered 1 to 6 are rolled. What is the probability that the sum of the rolls is 8?
Ans: Start with the total number of possible outcomes (the denominator). For this calculation you can use combinatorics. Notice that rolling two number cubes is like rolling cube 1 AND rolling cube 2. For each of these rolls, there are 6 possible outcomes (the numbers 1 to 6). Since AND equals multiply, there are 6 × 6 = 36 possible outcomes. This is the denominator of your fraction.

Next, figure out how many of those 36 possible rolls represent the desired outcome (a sum of 8). Can you think of an appropriate combinatorics formula? Probably not. Truth be told, the writers of this guide can't think of a formula either.

Fortunately, you don't need a formula. Only a limited number of combinations would work. Count them up! When you have to count, you never have to count too much. If the first die turns up a 1, the other die would need to roll a 7. This isn't possible, so eliminate that possibility. Keep counting; here are the rolls that work:
2 − 6
3 − 5
4 − 4
5 − 3
6 − 2
That's it; there are 5 combinations that work. Therefore, the probability of a sum of 8 is 5/36.

More Than One Event: “AND” vs. “OR”

• In both combinatorics and probability, there's a strong correlation between the concepts of "AND" and "OR" and mathematical operations. Specifically, in these contexts, "AND" signifies multiplication, while "OR" denotes addition.
• When dealing with probabilities or combinatorics problems involving multiple events or outcomes occurring together (i.e., in conjunction, represented by "AND"), the mathematical operation to find the combined probability or total number of outcomes is multiplication. For instance, if you want to find the probability of event A AND event B happening, you multiply the individual probabilities of A and B.
• On the other hand, when dealing with scenarios where you're considering the probability or count of either one event happening OR another (i.e., in isolation or in combination, represented by "OR"), the mathematical operation involved is addition. For example, if you want to find the probability of event A OR event B occurring, you add their individual probabilities.
• Understanding these connections between the logical connectors "AND" and "OR" in everyday language and their mathematical interpretations in combinatorics and probability allows for clearer problem-solving approaches. It essentially translates real-world scenarios into mathematical operations, simplifying the calculation of probabilities or the counting of outcomes in various situations.

Example 1: There is a 1/2 probability that a certain coin will turn up heads on any given toss.
What is the probability that two tosses of the coin will yield heads both times?
Ans: Calculate the probability that the coin lands on heads on the first flip AND heads on the second flip. The probability of heads on the first flip is 1/2. The probability of heads on the second flip is also 1/2. Since AND means multiply, the probability is

Example 2: The weather report for today states that there is a 40% chance of sun, a 25% chance of rain, and a 35% chance of hail. Assuming only one of the three outcomes can happen, what is the probability that it rains or hails today?
Ans: The question is asking for the probability of rain OR hail. Therefore, the probability is 25% + 35% = 60%. The calculation would change if both rain and hail can happen, but don't worry about that for now.

P(A) + P(Not A) = 1

• The equation P(A) + P(Not A) = 1 simplifies a fundamental probability principle: the likelihood of an event occurring plus the likelihood of it not occurring equals certainty. In simpler terms, the probability of something happening and the probability of it not happening combined must always equal 100%.
• Consider the scenario of rain or no rain to illustrate this principle. If there's a 25% chance of rain, then there's automatically a 75% chance that rain won't occur. This makes sense because when you add the probability of rain (25%) to the probability of no rain (75%), the total likelihood covers all possibilities, resulting in complete certainty, or 100%.
• Ultimately, this principle highlights that in any given situation, the sum of the likelihood of an event occurring and the likelihood of its opposite outcome happening accounts for all potential outcomes, ensuring that the total probability amounts to certainty.

Example: A person has a 40% chance of winning a game every time he or she plays it. If there are no ties, what is the probability that Asha loses the game the first time she plays and wins the second time she plays?
Ans: If the probability of winning the game is 40%, then the odds of not winning the game (losing) are 100% − 40% = 60%. Calculate the odds that Asha loses the game the first time AND wins the game the second time:
(60%) × (40%) = 0.6 × 0.4 = 0.24
The probability is 0.24, or 24%

The 1 − x Probability Trick

Suppose that a salesperson makes 5 sales calls, and you want to find the likelihood that he or she makes at least 1 sale. If you try to calculate this probability directly, you will have to confront 5 separate possibilities that constitute “success”: exactly 1 sale, exactly 2 sales, exactly 3 sales, exactly 4 sales, or exactly 5 sales. This would almost certainly be more work than you can reasonably do in two minutes. 

There is, however, another option. Instead of calculating the probability that the salesperson makes at least one sale, you can calculate the probability that the salesperson does not make at least one sale. Then, you could subtract that probability from 1. This shortcut works because the thing that does not happen represents a smaller number of the possible outcomes: that is, not getting at least 1 sale is the same thing as getting 0 sales, which is just one of the total possible outcomes. By contrast, making at least 1 sale represents 5 separate possible outcomes. When this occurs, it is much easier to calculate the probability for that one possible outcome (0 sales) and then subtract from 1. 

For complicated probability problems, decide whether it is easier to calculate the probability you want or the probability you do not want.

Example: A bag contains equal numbers of red, green, and yellow marbles. If Geeta pulls three marbles out of the bag, replacing each marble after she picks it, what is the probability that at least one will be red?
Ans: The quick way to answer this question is to calculate the probability that none of the marbles are red. For each of the three picks, there is a 2/3 probability that the marble will not be red. The probability that all three marbles will not be red is
If the probability that none of the marbles is red is 8/27, then the probability that at least one marble is red is
If you need to calculate the probability of an event (P(A)), there are two ways to calculate the probability:
P(A) or 1 − P(Not A)
When the question includes at least or at most language, the 1 − P(Not A) method is usually faster.

Solved Example

Example: What is the probability of getting at least 2 heads on 5 coin tosses?
Ans: Here is the more efficient way to solve this question without writing every possibility out. For every coin that you toss there are 2 possibilities. You can think of the total possibilities like a permutation problem.
__2__    __2__   __2__   __2__  __2_
1st             2nd          3rd         4th         5th
Just like in a GMAT permutations question when we are trying to determine the total number of codes possible or 4-digit numbers, we would multiply these individual probabilities together. Therefore, there are 2 x 2 x 2 x 2 x 2 = 2^5 = 32 total possibilities.
Next, we need to look at the numerator  (desired outcomes). We want to find the all of the possibilities that have at least 2 heads, which means that we could have 2 heads, 3 heads, 4 heads, or 5 heads. To do so, we would need to count all of the different ways that these possibilities could be arranged. Again, we find ourselves in a situation that will be time-consuming and fraught with error. Instead of going down this path, remember that the sum of the probabilities of a complete set of mutually exclusive possible outcomes is 1. Thus, as is often the case on “at least” probability questions, we can look for those options that are restricted. Then we only have to count the options that have 1 or 0 heads.
TTTTT
HTTTT
THTTT
TTHTT
TTTHT
TTTTH
There are only 6 of those, instead of the 26 possibilities the other way.
Finally, we can either subtract 6/32 from 1 in order to remove all of the restricted possibilities from 1 or we can subtract 6 from 32 and use the result as the desired possibilities. Either way, the answer is 26/32, which you can reduce down to 13/16.