Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Count distinct arrangements when some objects repeat
Use permutations and combinations to compute probabilities of compound events and solve problems.
Repeated objects make the all-labeled factorial overcount visible arrangements. We’ll inventory the total letters and each repeated group, start with all positional orders, divide by the internal permutations of identical …
Preview problemCount combinations when choosing a group and order does not matter
Use permutations and combinations to compute probabilities of compound events and solve problems.
A committee is determined by membership rather than listing order. We’ll classify the outcome as an unordered group, apply the combination formula, cancel shared factorial factors, evaluate the remaining quotient, …
Preview problemDecide whether a counting situation uses a permutation or a combination
Use permutations and combinations to compute probabilities of compound events and solve problems.
The key decision is whether rearranging the same selected people creates a new outcome. We’ll test that question for a committee with no assigned offices, conclude that membership alone defines …
Preview problemUse the multiplication principle to count outcomes in a multi-step choice situation
Use permutations and combinations to compute probabilities of compound events and solve problems.
A complete outfit requires one choice from every independent category. We’ll count possibilities stage by stage, multiply shirt and pants combinations, extend each partial outfit through all shoe choices, and …
Preview problemCount possible outcomes with restrictions using the multiplication principle, factorials, or combinations
Use permutations and combinations to compute probabilities of compound events and solve problems.
Restrictions can make different positions have different numbers of available digits. We’ll handle the nonzero first position first, restore zero to the pool for later positions, decrease the count after …
Preview problemFind the probability of an ordered outcome using permutation counts
Use permutations and combinations to compute probabilities of compound events and solve problems.
An exact code lives in an ordered sample space, because changing positions creates a different outcome. We’ll count all no-repeat codes with a descending product, express that total as a …
Preview problemFind the probability of an unordered selection using combinations
Use permutations and combinations to compute probabilities of compound events and solve problems.
A card hand is an unordered group, so numerator and denominator should use the same combination model. We’ll count all five-card hands, build favorable hands from the required hearts and …
Preview problemFind the probability of at least one desired outcome using the complement
Use permutations and combinations to compute probabilities of compound events and solve problems.
At least one is often simplest through the single way the event can fail. We’ll name the no-heads complement, use independence to multiply the four tail probabilities, subtract that result …
Preview problemFind the probability of getting an exact category breakdown in an unordered selection
Use permutations and combinations to compute probabilities of compound events and solve problems.
An exact suit breakdown is built from simultaneous category requirements inside an unordered hand. We’ll count the required heart groups and club groups separately, multiply them to form favorable hands, …
Preview problemFind the probability of a compound selection event using permutations or combinations
Use permutations and combinations to compute probabilities of compound events and solve problems.
A required rank in each draw position makes this an ordered without-replacement event. We’ll count all ordered three-card draws with decreasing deck totals, count favorable sequences by the available cards …
Preview problemEvaluate two explicit counting strategies for equivalence
Use permutations and combinations to compute probabilities of compound events and solve problems.
Two probability strategies can be compared only after fixing a common sample space. We’ll evaluate the disjoint direct cases, independently count and subtract the no-ace complement, place each favorable count …
Preview problemDecide whether a counting setup overcounts or undercounts
Use permutations and combinations to compute probabilities of compound events and solve problems.
To diagnose a counting error, compare what the formula treats as distinct with what the situation treats as distinct. We’ll interpret the permutation’s ordered positions, test whether a plain committee …
Preview problemDecode a combination or permutation probability expression
Use permutations and combinations to compute probabilities of compound events and solve problems.
A probability expression can be decoded from the denominator outward. We’ll identify the total outcome type and whether order matters, translate each numerator factor into a category count, check that …
Preview problemCompute participant probabilities and classify a selection method's fairness
Use probability to make fair decisions, such as lotteries or random selection.
Randomness alone does not establish fairness; the participant probabilities must be compared. We’ll map slips to students, find the probability of each equally likely slip, transfer that probability through the …
Preview problemCompute spinner allocations and classify fairness
Use probability to make fair decisions, such as lotteries or random selection.
Equal-looking spinner pieces matter only after their player assignments are counted. We’ll verify the sectors have equal area, map every sector to its player, total each player’s share of the …
Preview problemDecide whether a lottery-style drawing is fair by comparing each participant's probability of being selected
Use probability to make fair decisions, such as lotteries or random selection.
A one-winner drawing can still be fair when every participant has the same chance before the draw. We’ll count the identical slips, map one entry to each person, compute a …
Preview problemScreen simulation results for consistency with fairness using a stated rule
Use probability to make fair decisions, such as lotteries or random selection.
Simulation results should be judged by the stated screen, not by demanding exact equality. We’ll establish the fair benchmark, compute every absolute percentage-point deviation, retain the largest deviation, check both …
Preview problemCompare selection methods by fairness and expected attempts
Use probability to make fair decisions, such as lotteries or random selection.
Fairness and efficiency are separate comparisons. For each die method, we’ll trace the eventual probability assigned to every student, account for any reroll through conditional acceptance, test equality of participant …
Preview problemDecide whether a multi-stage random selection process is fair by comparing each participant's overall probability
Use probability to make fair decisions, such as lotteries or random selection.
Equal probabilities at the group stage do not automatically mean equal probabilities for individuals. We’ll trace one named person through each group branch, multiply the group chance by the within-group …
Preview problemInterpret fairness by naming the unit and checking equal probabilities
Use probability to make fair decisions, such as lotteries or random selection.
A fairness claim must name the unit receiving equal chances. We’ll use the student as that unit, map each student to the equally likely slips, compute one student’s probability, show …
Preview problemIdentify the hidden bias in a proposed random selection method by comparing probabilities
Use probability to make fair decisions, such as lotteries or random selection.
Equally likely slips can still create unequal student chances when names appear different numbers of times. We’ll count each student’s entries, include the duplicate in the total, form probabilities from …
Preview problemFind the expected value of a game from its outcomes and probabilities
Analyze decisions and strategies with probability concepts in applied settings.
Expected value is a probability-weighted average of signed net outcomes. We’ll record gains as positive and losses as negative, confirm the probabilities cover the game, multiply each outcome by its …
Preview problemInterpret expected value as a long-run average with a stated unit
Analyze decisions and strategies with probability concepts in applied settings.
Interpreting expected value requires preserving its sign, quantity, and per-trial unit. We’ll translate the negative net amount into the player’s gain-or-loss direction, connect it to total result divided by many …
Preview problemClassify a game as fair, favorable, or unfavorable using expected value
Analyze decisions and strategies with probability concepts in applied settings.
Game classification comes from comparing the expected net result with zero. We’ll keep gain and loss signs attached, verify the outcome probabilities total one, compute each probability-weighted contribution, add them, …
Preview problemChoose the better strategy by comparing expected values
Analyze decisions and strategies with probability concepts in applied settings.
A strategy comparison must follow the criterion named in the question. We’ll place both expected values in the same dollars-per-use unit, apply the greater-long-run-payoff rule, compare the values and their …
Preview problemFind the overall probability that a strategy succeeds
Analyze decisions and strategies with probability concepts in applied settings.
The strategy’s success event requires both independent stages, not merely one of them. We’ll define the joint event, verify that the first result does not change the second-stage rate, apply …
Preview problemChoose the lower expected monetary cost between coverage and no coverage
Analyze decisions and strategies with probability concepts in applied settings.
A possible large loss should be probability-weighted before it is compared with a certain premium. We’ll model both no-plan outcomes, compute their expected monetary cost, record the plan’s fixed cost …
Preview problemCompute strategy expected values and choose under an explicit criterion
Analyze decisions and strategies with probability concepts in applied settings.
An action decision under a maximize-expected-value rule begins with signed net outcomes. We’ll weight the gain and loss by their probabilities, add those contributions, model declining as a certain zero, …
Preview problemCompare simulated strategy averages and variability under a stated screen
Analyze decisions and strategies with probability concepts in applied settings.
A simulation screen and a variability claim require different evidence. We’ll compute the observed mean gap, test the sample size and gap against their separate thresholds, classify only what that …
Preview problemEvaluate whether a strategy claim is supported by expected value
Analyze decisions and strategies with probability concepts in applied settings.
A profitability claim translates into a sign test against the break-even benchmark of zero. We’ll preserve the expected value’s negative sign and per-play unit, compare it with zero, interpret the …
Preview problem