Probability
50 flashcards covering Probability for the ACT Math section.
Probability is a mathematical concept that measures the likelihood of an event occurring, expressed as a number between 0 and 1, where 0 means it's impossible and 1 means it's certain. For instance, if you roll a six-sided die, the probability of getting a 3 is 1 in 6, or about 0.167. It helps us make sense of uncertainty in everyday situations, like predicting weather or game outcomes, and it's foundational for fields like statistics.
On the ACT Math section, probability appears in questions that test basic calculations, such as finding the chance of drawing a specific card from a deck or the probability of multiple events happening together. Common traps include mixing up "and" (intersection) with "or" (union) events, or forgetting to account for dependent versus independent scenarios. Focus on understanding key formulas, like the basic probability rule, and practicing word problems that involve combinations or permutations to avoid errors.
A good tip: Always check if events are mutually exclusive before adding probabilities.
Terms (50)
- 01
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1, where 0 means impossible and 1 means certain.
- 02
Sample space
The sample space is the set of all possible outcomes of a random experiment.
- 03
Event
An event is a subset of the sample space, representing one or more outcomes that may occur in a probability experiment.
- 04
Outcome
An outcome is a single possible result of a random experiment, such as rolling a 6 on a die.
- 05
Favorable outcome
A favorable outcome is a result that satisfies the conditions of the event being considered in a probability calculation.
- 06
Random experiment
A random experiment is a process that leads to one of several possible outcomes, each with some degree of uncertainty, like flipping a coin.
- 07
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 two coins.
- 08
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.
- 09
Mutually exclusive events
Mutually exclusive events are events that cannot occur at the same time, meaning if one happens, the others cannot.
- 10
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 occurring.
- 11
Multiplication rule for probability
The multiplication rule states that for independent events, the probability of both A and B occurring is the product of their individual probabilities.
- 12
Conditional probability
Conditional probability is the probability of an event occurring given that another event has already occurred, calculated as the probability of both events divided by the probability of the given event.
- 13
Permutations
Permutations are the number of ways to arrange a set of items in a specific order, calculated using the formula P(n, r) = n! / (n - r)!.
- 14
Combinations
Combinations are the number of ways to select items from a set without regard to order, calculated using the formula C(n, r) = n! / (r! × (n - r)!).
- 15
Factorial
Factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n, with 0! equaling 1.
- 16
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.
- 17
Probability of at least one
The probability of at least one event occurring is 1 minus the probability that none of the events occur.
- 18
Probability of at most one
The probability of at most one event occurring is the sum of the probability that none occur and the probability that exactly one occurs.
- 19
Expected value
Expected value is the long-run average value of repetitions of an experiment, calculated by summing each outcome multiplied by its probability.
- 20
Odds in favor
Odds in favor of an event are the ratio of the probability that the event will occur to the probability that it will not occur.
- 21
Simple probability formula
The simple probability of an event is the number of favorable outcomes divided by the total number of possible outcomes in the sample space.
- 22
Probability with replacement
Probability with replacement means items are put back after being selected, so the sample space remains the same for each draw.
- 23
Probability without replacement
Probability without replacement means items are not put back after being selected, so the sample space changes with each draw.
- 24
Union of events
The union of events A and B is the event that either A or B or both occur.
- 25
Intersection of events
The intersection of events A and B is the event that both A and B occur.
- 26
Disjoint events
Disjoint events are the same as mutually exclusive events, where the intersection has a probability of zero.
- 27
Inclusive events
Inclusive events are events that may overlap, meaning both can occur simultaneously.
- 28
Tree diagram
A tree diagram is a visual tool used to list all possible outcomes of a sequence of events, helping to calculate probabilities step by step.
- 29
Venn diagram for probability
A Venn diagram illustrates the relationships between events, showing overlaps to help visualize unions and intersections in probability problems.
- 30
Geometric probability
Geometric probability involves calculating the probability of an event based on ratios of lengths, areas, or volumes in a geometric context.
- 31
Theoretical probability
Theoretical probability is the likelihood of an event based on reasoning, calculated as the number of favorable outcomes over total possible outcomes.
- 32
Empirical probability
Empirical probability is the likelihood of an event based on observed frequency from experiments or data.
- 33
Random variable
A random variable is a variable that represents the outcome of a random experiment, taking on different values with associated probabilities.
- 34
Binomial probability
Binomial probability calculates the chance of getting exactly k successes in n independent trials, each with the same success probability.
- 35
Binomial coefficient
The binomial coefficient, denoted C(n, k), is the number of ways to choose k items from n without regard to order, used in binomial expansions.
- 36
Strategy for counting outcomes
To count outcomes accurately, list all possibilities systematically, use permutations for order-dependent problems, and combinations for order-independent ones.
- 37
Common trap: Overcounting
Overcounting occurs when arrangements are counted multiple times due to failing to account for identical items, leading to inflated probabilities.
- 38
Common trap: Confusing and/or
Confusing 'and' with 'or' in probability can lead to errors; 'and' requires multiplication for independent events, while 'or' uses the addition rule.
- 39
Common trap: Dependent vs. independent
Mistaking dependent events for independent ones can result in incorrect multiplication of probabilities.
- 40
Example: Coin flip probability
The probability of getting heads on a fair coin flip is 0.5, as there are two equally likely outcomes: heads or tails.
For two flips, the probability of two heads is 0.25.
- 41
Example: Die roll probability
The probability of rolling a 3 on a fair six-sided die is 1/6, since there is one favorable outcome out of six possible.
- 42
Example: Drawing a card
The probability of drawing an ace from a standard deck of 52 cards is 4/52, as there are four aces.
- 43
Example: Birthday probability
The probability that two people share a birthday in a group of 23 is about 0.5, illustrating the birthday paradox.
- 44
Example: Probability of union
If the probability of A is 0.4 and B is 0.3, and they overlap with P(A and B) of 0.1, then P(A or B) is 0.6.
- 45
Example: Conditional probability
If the probability of rain given clouds is 0.7, it means that when clouds are present, rain occurs 70% of the time.
- 46
Example: Permutations of letters
The number of ways to arrange the letters in 'ACT' is 6, since there are 3 distinct letters.
- 47
Example: Combinations of items
The number of ways to choose 2 books from 5 is 10, calculated as C(5, 2).
- 48
Example: Expected value of a game
In a game where you win $2 with probability 0.3 and lose $1 with probability 0.7, the expected value is 2(0.3) + (-1)(0.7) = 0.3.
- 49
Simulation in probability
Simulation involves using random trials, like computer-generated numbers, to estimate probabilities when exact calculations are complex.
- 50
Law of large numbers
The law of large numbers states that as the number of trials increases, the experimental probability approaches the theoretical probability.