Probability basics
55 flashcards covering Probability basics for the SAT Math section.
Probability is a fundamental concept that measures the likelihood of an event occurring, expressed as a number between 0 and 1. For instance, if you roll a fair six-sided die, the probability of getting a 3 is 1 out of 6, or about 0.167. It involves basic ideas like possible outcomes, sample spaces, and simple calculations for independent or dependent events. Understanding these basics helps you make sense of uncertainty in everyday situations and lays the groundwork for more advanced math.
On the SAT Math section, probability questions typically involve calculating chances in scenarios like coin flips, card draws, or surveys, often through word problems or multiple-choice formats. Common traps include confusing independent events with dependent ones, overlooking total possibilities, or misapplying formulas like the addition or multiplication rules. Focus on mastering definitions, practicing quick calculations, and interpreting questions carefully to avoid errors.
Remember to always draw a diagram of the sample space for complex problems.
Terms (55)
- 01
Probability
Probability measures the likelihood of an event occurring, calculated as the number of favorable outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.
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Sample space
The sample space is the set of all possible outcomes of a probability experiment, such as the numbers 1 through 6 for a fair six-sided die.
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Event
An event is a subset of the sample space, representing one or more outcomes that may occur, like rolling an even number on a die.
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Outcome
An outcome is a single result from a probability experiment, such as getting heads on a coin flip.
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Favorable outcome
A favorable outcome is an outcome that meets the criteria of the event being considered, like drawing a heart from a deck of cards.
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Equally likely outcomes
Equally likely outcomes are results in a sample space that each have the same probability of occurring, such as the six faces of a fair die.
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Theoretical probability
Theoretical probability is the calculated probability based on the ratio of favorable outcomes to total possible outcomes, without performing the experiment.
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Experimental probability
Experimental probability is the probability estimated from actual trials of an experiment, calculated as the number of successful trials divided by the total number of trials.
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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, such as flipping a coin and rolling a die.
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Dependent events
Dependent events are two or more events where the outcome of one affects the probability of the other, such as drawing cards without replacement from a deck.
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Mutually exclusive events
Mutually exclusive events are events that cannot occur at the same time, so the probability of both happening is zero, like rolling a 1 or a 6 on a single die roll.
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Not mutually exclusive events
Not mutually exclusive events are events that can occur at the same time, requiring the addition rule with subtraction for overlaps, like drawing a heart or a king from a deck.
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Addition rule for probability
The addition rule states that the probability of A or B occurring is the probability of A plus the probability of B minus the probability of both A and B, used for any two events.
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Multiplication rule for independent events
For independent events, the probability of both A and B occurring is the product of their individual probabilities, such as the chance of two heads in separate coin flips.
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Multiplication rule for dependent events
For dependent events, the probability of both A and B is the probability of A times the probability of B given A has occurred, like drawing two aces from a deck without replacement.
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Complementary event
The complementary event of A is the event that A does not occur, and its probability is 1 minus the probability of A, often easier to calculate for 'at least one' scenarios.
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Probability of at least one
The probability of at least one event occurring is 1 minus the probability of none occurring, useful for problems like getting at least one head in three coin flips.
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Permutations
Permutations are the number of ways to arrange items in a specific order, calculated using factorials or the formula P(n, r) = n! / (n-r)!, often used in probability for ordered outcomes.
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Combinations
Combinations are the number of ways to select items without regard to order, calculated as C(n, r) = n! / (r! (n-r)!), used for probability when order doesn't matter.
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Factorial
Factorial of a number n, denoted n!, is the product of all positive integers from 1 to n, essential for calculating permutations and combinations in probability.
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With replacement
In probability experiments, with replacement means an item is returned to the set after selection, keeping subsequent probabilities the same, like drawing marbles from a bag and putting them back.
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Without replacement
Without replacement means an item is not returned after selection, altering the probabilities for subsequent draws, such as picking cards from a deck.
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Simple event
A simple event is a single outcome from the sample space, like rolling a specific number on a die, as opposed to a combination of outcomes.
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Compound event
A compound event is an event made up of two or more simple events, such as rolling an even number or greater than 4 on a die.
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Impossible event
An impossible event is one that cannot occur, with a probability of 0, like rolling a 7 on a standard six-sided die.
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Certain event
A certain event is one that is guaranteed to occur, with a probability of 1, such as rolling a number between 1 and 6 on a standard die.
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Expected value
Expected value is the long-run average value of repetitions of an experiment, calculated by summing each outcome multiplied by its probability, used in decision-making problems.
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Geometric probability
Geometric probability involves finding the probability based on areas or lengths, such as the chance a randomly chosen point in a square falls inside a circle inscribed in it.
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Inclusion-exclusion principle
The inclusion-exclusion principle calculates the probability of the union of two or more events by adding and subtracting overlapping probabilities to avoid double-counting.
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Venn diagrams in probability
Venn diagrams visually represent events and their overlaps to help calculate probabilities of unions and intersections, especially for not mutually exclusive events.
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Tree diagrams
Tree diagrams map out all possible outcomes of sequential events, showing probabilities at each branch, useful for problems with dependent events.
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Odds
Odds express the ratio of the probability of an event occurring to it not occurring, such as 3 to 1 odds meaning three favorable outcomes for every one unfavorable.
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Converting odds to probability
To convert odds of a to b into probability, use the formula: probability = a / (a + b), where a is for the event and b is against it.
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Strategy for counting outcomes
A strategy for counting outcomes involves systematically listing or using formulas like permutations and combinations to ensure all possibilities are covered without omission.
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Avoiding double-counting
Avoiding double-counting in probability means carefully accounting for overlaps when adding probabilities or counting outcomes, often using inclusion-exclusion.
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When to use permutations
Use permutations when the order of selection matters in probability problems, such as arranging books on a shelf.
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When to use combinations
Use combinations when the order of selection does not matter, like choosing a committee from a group.
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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, as in drawing cards without replacement.
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Common trap: Misadding probabilities
Misadding probabilities often occurs when forgetting to subtract overlaps for not mutually exclusive events, resulting in an overestimation of the union.
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Probability of rolling a 6 on a die
The probability of rolling a 6 on a fair six-sided die is 1/6, as there is one favorable outcome out of six possible ones.
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Probability of two heads in coin flips
For two independent coin flips, the probability of both being heads is 1/2 times 1/2, which equals 1/4.
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Probability of drawing a red card
The probability of drawing a red card from a standard deck is 26/52, or 1/2, since half the cards are red.
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Probability of at least one head in two flips
The probability of at least one head in two coin flips is 1 minus the probability of no heads, which is 1 - 1/4 = 3/4.
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Probability of two aces from a deck
The probability of drawing two aces without replacement from a standard deck is (4/52) times (3/51), approximately 0.0045.
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Probability in urn problems
In urn problems, probability is calculated based on the number of desired items over total items, adjusted for replacement or multiple draws.
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Basic probability formula
The basic probability formula is P(event) = number of favorable outcomes / total number of possible outcomes, assuming equally likely outcomes.
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Probability of unions
The probability of the union of events A and B is P(A) + P(B) - P(A and B), ensuring overlaps are not double-counted.
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Probability of intersections
The probability of the intersection of events, or both occurring, depends on independence; for independent events, it's the product of their probabilities.
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Overlapping sets in probability
Overlapping sets represent events that can occur together, requiring subtraction of the intersection when calculating the union's probability.
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Example: Birthday problem
In the birthday problem, the probability that at least two people in a group share a birthday is calculated using complementary probability, often surprisingly high for small groups.
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Trap: Overcounting in probability
Overcounting occurs when listing outcomes fails to account for duplicates, leading to incorrect probabilities, so use systematic methods like combinations.
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Conditional probability
Conditional probability is the probability of an event given that another event has occurred, calculated as P(A and B) / P(B) for event A given B.
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Binomial probability
Binomial probability calculates the chance of exactly k successes in n independent trials, using the formula C(n, k) times p^k times (1-p)^(n-k), for events like coin flips.
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Probability distributions
Probability distributions describe the likelihood of each outcome in a sample space, such as a uniform distribution for a fair die.
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Strategy for word problems
For probability word problems, identify the sample space, determine if events are independent or dependent, and apply the appropriate rules to calculate the required probability.