Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
Page 18 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Estimate the percent within 1, 2, or 3 standard deviations of the mean using the empirical rule
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Translate “within” into a central interval extending the same number of standard deviations on both sides of the mean. Then match that full band to the empirical rule: approximately 68, …
Preview problemFind the z-score of a value given the mean and standard deviation
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
A z-score converts raw distance from the mean into standard-deviation units. Subtract the mean from the value in that order, then divide by the positive standard deviation. The sign tells …
Preview problemInterpret z-score direction and absolute distance
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Separate the two jobs of a z-score. Its sign gives direction from the mean—positive above, negative below, and zero at the mean—while its absolute value gives a nonnegative distance measured …
Preview problemCompute a normal area to four decimals
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Sketch the requested normal region conceptually before entering technology so the inequality determines the correct lower and upper bounds. Supply those bounds first, followed by the mean and positive standard …
Preview problemFind a normal cutoff to one decimal
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Inverse normal starts with cumulative area to the left of the desired cutoff. Convert percentile or top-tail language into that lower-tail probability, then enter area, mean, and positive standard deviation …
Preview problemCompare which value is relatively higher in its own normal distribution using z-scores
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Raw distances from different means are not comparable when the distributions have different spreads. Standardize each observation using its own mean and standard deviation, then place both results on the …
Preview problemDecide whether a normal model is appropriate for a quantitative data set
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Judge a normal model by the distribution's overall shape and context, not by demanding perfect agreement. A roughly symmetric, single-mound quantitative distribution without strong outliers passes the main visual checks …
Preview problemEstimate a population percentage from a normal distribution using mean and standard deviation
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Locate each bound relative to the mean by measuring its distance in standard deviations. If the bounds form a familiar symmetric band around the mean, apply the matching empirical-rule percentage; …
Preview problemInterpret how changing the mean or standard deviation changes a normal model
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Keep the two parameters in separate mental boxes: one controls where a normal curve is centered, while the other controls how spread out it is. Track only the parameter named …
Preview problemList ordered path probabilities and test equal-outcome fairness
Use probabilities to make fair decisions in more complex settings.
For a multistage chance process, the basic objects are complete ordered paths, not the outcomes from either stage by itself. Multiply the branch probabilities along one path, then repeat for …
Preview problemJudge simulated fairness with an explicit tolerance
Use probabilities to make fair decisions in more complex settings.
A simulation should be judged on the same scale as its fairness rule, so convert every count to a proportion first. Compare each observed proportion with its equal-chance target using …
Preview problemIdentify and test the hidden equiprobability assumption
Use probabilities to make fair decisions in more complex settings.
A list of named results is not automatically a list of equally likely outcomes. Trace each name back to the genuinely equiprobable atomic outcomes, then count how many atoms feed …
Preview problemCompare participant probabilities under an explicit fairness rule
Use probabilities to make fair decisions in more complex settings.
Fairness belongs to participant win chances, not simply to the number of participants or to a probability total of one. When the underlying outcomes are equally likely, turn each participant's …
Preview problemMap equiprobable outcomes to equal participant chances
Use probabilities to make fair decisions in more complex settings.
Design fairness by distributing probability mass, not just by writing participant names. With equal sectors, equal probability mass comes from equal sector counts, so first test whether the total divides …
Preview problemAccount for stage-dependent win paths
Use probabilities to make fair decisions in more complex settings.
A win can be reached through more than one complete branch, and each branch must carry the probability of reaching it. Multiply within a path, then add the disjoint paths …
Preview problemCount favorable outcomes for the correct fairness unit
Use probabilities to make fair decisions in more complex settings.
Start the audit at the smallest equally likely unit: here, each ticket is one atomic chance to win. Group those units by owner and divide each owner's count by the …
Preview problemRepair an unfair random method with exact probabilities
Use probabilities to make fair decisions in more complex settings.
A reroll repair separates a single spin from the eventual completed trial. First make the accepted probability mass equal across participants, then condition on the event that a participant was …
Preview problemAudit chance, efficiency, and transparency separately
Use probabilities to make fair decisions in more complex settings.
Treat chance, efficiency, and transparency as three separate audit questions. Equal participant probability answers the fairness question; acceptance probability and expected attempts measure operational efficiency; visible, verifiable rules support transparency. …
Preview problemChoose among fair methods using stated criteria
Use probabilities to make fair decisions in more complex settings.
When criteria are ordered, treat the first one as a gate rather than blending everything into one score. Remove any method that fails exact fairness, even if it looks efficient. …
Preview problemFind the expected value by multiplying each outcome by its probability and adding
Analyze decisions and strategies using probability concepts in more complex settings.
Expected value is a probability-weighted balance of every possible payoff. Keep gains positive and losses negative, multiply each signed outcome by the probability attached to it, and add the contributions. …
Preview problemChoose between options using expected value
Analyze decisions and strategies using probability concepts in more complex settings.
A comparison by expected value is meaningful only when both quantities use the same money and time units. Compare the complete decimal amounts, and use their difference to describe the …
Preview problemFind the net expected value by weighting outcomes and then including a fixed cost
Analyze decisions and strategies using probability concepts in more complex settings.
Separate the random payout from the fixed cost so neither is counted incorrectly. First average the possible payouts with their probabilities, then subtract the cost that occurs on every play …
Preview problemCompare expected value and defined downside-risk metrics
Analyze decisions and strategies using probability concepts in more complex settings.
Expected value compresses a payoff distribution into one average, so equal averages can hide very different experiences. After computing each mean, return to the actual outcomes and measure downside relative …
Preview problemCompare simulated strategies on return, downside, and reliability
Analyze decisions and strategies using probability concepts in more complex settings.
Read a simulation comparison in separate columns: average return, frequency of loss, severity of loss, and reliability of the estimates. One strategy can lead on return while another leads on …
Preview problemCompare insurance expected cost separately from risk protection
Analyze decisions and strategies using probability concepts in more complex settings.
Put the protected and unprotected costs on the same time horizon before comparing them. The unprotected side uses a probability-weighted loss, while the protection price is a certain cost. Any …
Preview problemUpdate a state probability after new information
Analyze decisions and strategies using probability concepts in more complex settings.
Updating after evidence means restricting attention to every route that could have produced what was observed. Multiply each prior state probability by that state's evidence likelihood to get joint branch …
Preview problemInterpret false positives with base rates
Analyze decisions and strategies using probability concepts in more complex settings.
Base-rate problems become clearer when percentages are converted to expected counts in the stated population. Split the population by condition status first, then apply sensitivity only to the condition branch …
Preview problemUse conditional probabilities in an expected-value decision
Analyze decisions and strategies using probability concepts in more complex settings.
Once new information supplies state probabilities, use those same conditional probabilities for every alternative. Weight each alternative's payoff in every state, including losses, and compare the resulting expectations under the …
Preview problemApply explicit downside constraints before optimizing
Analyze decisions and strategies using probability concepts in more complex settings.
A constrained decision has two stages, and reversing them changes the problem. First compute the downside measure named in the rule and remove every alternative that crosses the hard limit. …
Preview problem