Probability on the GMAT
55 flashcards covering Probability on the GMAT for the GMAT Quantitative section.
Probability is the measure of how likely an event is to occur, expressed as a fraction or percentage between 0 and 1. For example, if you flip a fair coin, the probability of getting heads is 1 out of 2 possible outcomes, or 0.5. It involves calculating the ratio of favorable results to the total possible results, and it often deals with concepts like independent events, where one outcome doesn't affect another, or dependent events, which do. On the GMAT, understanding probability helps you tackle problems that test logical reasoning and decision-making under uncertainty, which are key for business and management scenarios.
In the Quantitative section of the GMAT, probability questions typically appear as problem-solving items, often combined with permutations, combinations, or conditional probability. Common traps include miscounting outcomes, confusing mutually exclusive events, or overlooking dependencies between events, which can lead to incorrect answers. Focus on mastering basic formulas, practicing word problems that describe real-world scenarios, and double-checking your calculations to avoid careless errors.
A concrete tip: Always draw a diagram or list outcomes to visualize the problem clearly.
Terms (55)
- 01
Probability
Probability is a measure of the likelihood that a specific event will occur, calculated as the number of favorable outcomes divided by the total number of possible outcomes in the sample space.
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Sample Space
The sample space is the set of all possible outcomes of a random experiment, which serves as the foundation for calculating probabilities.
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Event
An event is a subset of the sample space, representing one or more outcomes that may occur during a random experiment.
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Favorable Outcomes
Favorable outcomes are the specific results from the sample space that satisfy the conditions of the event being considered in a probability calculation.
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Basic Probability Formula
The basic formula for probability is P(event) = number of favorable outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.
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Independent Events
Independent events are two or more events where the occurrence of one does not affect the probability of the other occurring.
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Mutually Exclusive Events
Mutually exclusive events are events that cannot occur at the same time, meaning if one happens, the others cannot.
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Addition Rule for Probability
The addition rule states that for two events, the probability of either occurring is the sum of their individual probabilities minus the probability of both occurring.
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Multiplication Rule for Independent Events
For independent events, the multiplication rule calculates the probability of both occurring as the product of their individual probabilities.
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Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred, calculated as P(A|B) = P(A and B) / P(B).
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Complementary Events
Complementary events are a pair where one is the opposite of the other, and their probabilities sum to 1, allowing easier calculation of the complement.
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Union of Events
The union of events is the event that at least one of the events occurs, and its probability can be found using the addition rule.
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Intersection of Events
The intersection of events is the event that all specified events occur simultaneously, often calculated using the multiplication rule.
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Dependent Events
Dependent events are events where the outcome of one affects the probability of the other, requiring adjustment in calculations like conditional probability.
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At Least One
The probability of at least one event occurring is calculated as 1 minus the probability of none of the events occurring, useful for scenarios like multiple trials.
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Exactly K Successes
Exactly K successes refers to the probability of achieving a specific number of successes in a fixed number of trials, often involving binomial probability.
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Combinations
Combinations are the number of ways to select items from a group without regard to order, calculated as C(n, k) = n! / (k! (n - k)!), essential for probability problems.
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Permutations
Permutations are the number of ways to arrange items in a specific order, calculated as P(n, k) = n! / (n - k)!, used when sequence matters in probability.
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Factorial
Factorial of a number n, denoted n!, is the product of all positive integers from 1 to n, a key component in formulas for combinations and permutations.
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Binomial Probability
Binomial probability calculates the chance of exactly k successes in n independent trials, each with the same success probability, using the formula C(n, k) p^k (1-p)^(n-k).
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Expected Value
Expected value is the long-run average value of repetitions of an experiment, calculated as the sum of each outcome multiplied by its probability.
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Geometric Probability
Geometric probability involves finding the probability of an event based on ratios of lengths, areas, or volumes, often in continuous uniform distributions.
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Probability with Replacement
In probability with replacement, items are returned to the pool after selection, keeping probabilities constant for each draw in sequential events.
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Probability Without Replacement
In probability without replacement, items are not returned after selection, so the total number of outcomes decreases with each draw.
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Joint Probability
Joint probability is the probability of two or more events occurring together, represented as P(A and B), and is used in more complex event calculations.
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Marginal Probability
Marginal probability is the probability of an event regardless of other variables, often derived from joint probabilities in tables or matrices.
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Law of Total Probability
The law of total probability breaks down the probability of an event by considering all possible mutually exclusive scenarios that could lead to it.
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Common Trap: Assuming Independence
A common trap is assuming events are independent when they are not, leading to incorrect multiplication of probabilities without verifying dependence.
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Strategy for Counting Problems
For counting problems in probability, systematically list or calculate the total outcomes and favorable ones, using tools like combinations to avoid overcounting.
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Probability of No Events
The probability of no events occurring is the complement of the probability of at least one occurring, useful in problems involving failures or absences.
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Dice Probability Example
In dice problems, calculate probabilities based on the number of faces, such as the probability of rolling a sum of 7 with two dice being 6 out of 36 possible outcomes.
For two six-sided dice, the outcomes summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
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Coin Flip Probability
Coin flip probability assumes a fair coin, where the chance of heads or tails is 0.5, and multiple flips are independent events.
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Card Drawing Probability
Card drawing probability considers a standard 52-card deck, where events like drawing a heart depend on whether cards are replaced.
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Marbles in a Bag
In marbles problems, probability is based on the ratio of desired marbles to total marbles, adjusting for draws without replacement.
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Birthday Problem
The birthday problem calculates the probability that in a group, at least two people share a birthday, illustrating counterintuitive aspects of probability.
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Probability Trees
Probability trees are diagrams that map out sequential events and their probabilities, helping visualize and calculate combined outcomes.
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Venn Diagrams for Events
Venn diagrams visually represent events and their overlaps, aiding in calculating unions and intersections in probability problems.
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Odds vs. Probability
Odds are the ratio of the probability of an event occurring to it not occurring, while probability is the direct likelihood, and they can be converted between.
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Calculating Odds
To calculate odds, divide the number of favorable outcomes by the number of unfavorable outcomes, such as 3:2 odds for an event with probability 3/5.
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Expected Value Calculation
Expected value is computed by multiplying each possible outcome by its probability and summing them, guiding decisions in risk scenarios.
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Variance in Probability
Variance measures the spread of possible outcomes around the expected value, calculated as the expected value of the squared deviation from the mean.
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Standard Deviation Example
Standard deviation is the square root of variance, indicating the typical deviation from expected value, as in assessing risk in investments.
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Bernoulli Trial
A Bernoulli trial is a random experiment with two possible outcomes: success or failure, each with a fixed probability, forming the basis for binomial experiments.
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Binomial Coefficient
The binomial coefficient, denoted C(n, k), represents the number of ways to choose k successes in n trials, central to binomial probability calculations.
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Hypergeometric Distribution
Hypergeometric distribution calculates the probability of k successes in draws without replacement from a finite population, common in sampling problems.
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Strategy for Word Problems
For probability word problems, identify the sample space, define events clearly, and apply appropriate rules while watching for dependencies or exclusions.
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Common Error: Double-Counting
A common error is double-counting overlapping outcomes in unions, which is corrected by subtracting the intersection in the addition rule.
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Misapplying Multiplication Rule
Misapplying the multiplication rule occurs when treating dependent events as independent, leading to inflated probability estimates.
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Probability with And
Probability with 'and' means the joint occurrence of events, calculated via multiplication for independent events or conditional probability for dependent ones.
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Probability with Or
Probability with 'or' means at least one event occurs, calculated using the addition rule and accounting for any overlap between events.
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Trap: Ignoring Sample Space Size
A trap is ignoring changes in sample space size in without-replacement scenarios, which alters probabilities for subsequent events.
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Advanced: Conditional on Multiple Events
In advanced problems, conditional probability may depend on multiple prior events, requiring step-by-step application of the formula.
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Expected Value of a Game
The expected value of a game is the average payoff per play, helping determine if it's fair by comparing to zero or entry costs.
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Geometric Series in Probability
Geometric series can sum infinite probabilities, like in repeated independent trials until success, relating to geometric distribution.
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Strategy for At Least Problems
For 'at least' problems, calculate the complement (none occurring) and subtract from 1 to simplify computations with multiple events.