Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 20 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Classify dependence and identify the decisive probability evidence
Determine event independence using P(A and B)=P(A)P(B).
A direct independence test compares a conditional probability with its unconditional counterpart. We’ll write both exact fractions, cross-multiply rather than rely on rounded decimals, interpret their inequality as a probability …
Preview problemDecide whether a probability context suggests independent events and confirm with P(A and B)=P(A)P(B)
Determine event independence using P(A and B)=P(A)P(B).
Context can suggest independence when separate devices do not influence each other, but a numerical check confirms it. We’ll make the contextual prediction, use representative event probabilities, compute their product …
Preview problemFind a missing probability using independence and \(P(A and B) = P(A)P(B)\)
Determine event independence using P(A and B)=P(A)P(B).
Independence turns the joint probability into a product equation with one missing factor. We’ll substitute the known joint and marginal values, divide by the nonzero coefficient to isolate the unknown …
Preview problemDecide whether two events are independent by comparing P(A and B) to P(A)P(B)
Determine event independence using P(A and B)=P(A)P(B).
The reported joint-equals-product equation is exactly the probability criterion for independence. We’ll define the two contextual events, match the equality to that criterion, translate the classification into a statement about …
Preview problemFind \(P(A \mid B)\) from counts by dividing within the conditioned group
Understand conditional probability and connect independence to unchanged conditional probabilities.
The word given restricts the denominator to the conditioned group. We’ll identify that reduced universe, count the favorable overlap inside it, form favorable-over-conditioned rather than using the overall total, reduce …
Preview problemFind a conditional probability from P(A and B) and P(B)
Understand conditional probability and connect independence to unchanged conditional probabilities.
Conditional probability divides the joint probability by the probability of the conditioning event. We’ll place those quantities in their correct numerator-denominator roles, substitute the decimals, scale to an integer fraction, …
Preview problemInterpret conditional probability notation in words
Understand conditional probability and connect independence to unchanged conditional probabilities.
Conditional notation is directional: the event before the bar is measured inside the condition after the bar. We’ll read the bar as given, identify the restricted sample space, translate the …
Preview problemWrite a conditional probability statement in notation
Understand conditional probability and connect independence to unchanged conditional probabilities.
Writing a conditional requires preserving which event is measured and which event supplies the condition. We’ll place the target before the vertical bar, place the restricted group after it, use …
Preview problemCalculate and compare two reversed conditional probabilities
Understand conditional probability and connect independence to unchanged conditional probabilities.
Reversing a conditional keeps the overlap numerator but changes the conditioned-group denominator. We’ll form each ratio from its own condition, reduce both fractions, compare them on a common scale, and …
Preview problemDecide whether two events are independent by comparing a probability to its conditional probability
Understand conditional probability and connect independence to unchanged conditional probabilities.
With a positive conditioning probability, independence means conditioning on B leaves A’s probability unchanged. We’ll state that criterion, compare the conditional value with A’s marginal value, interpret equality as no …
Preview problemFind a joint probability from a conditional probability and the condition probability
Understand conditional probability and connect independence to unchanged conditional probabilities.
The conditional formula can be rearranged to recover the joint probability. We’ll start with joint over condition, multiply by the conditioning probability, substitute the supplied values, and divide the resulting …
Preview problemUse conditional probability for two draws without replacement
Understand conditional probability and connect independence to unchanged conditional probabilities.
Without replacement, the second-draw probability must reflect the changed deck. We’ll compute the first favorable probability, update both favorable and total counts after that outcome, multiply along the conditional path, …
Preview problemInterpret a conditional probability from Venn-diagram region counts
Understand conditional probability and connect independence to unchanged conditional probabilities.
Conditioning on B makes the entire B circle the new denominator, including its overlap and B-only regions. We’ll total that conditioned group, use the overlap as the favorable count for …
Preview problemJudge whether a claim is supported by conditional probability information
Understand conditional probability and connect independence to unchanged conditional probabilities.
A comparative claim is supported by conditional rates only when they measure the same outcome across the stated groups. We’ll align the outcome, compare the two group probabilities and their …
Preview problemComplete a two-way frequency table, including all margins
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
Completing a two-way table requires each margin to stay attached to the cells it summarizes. We’ll add across every row, add down every column, compute the grand total from both …
Preview problemComplete missing row totals, column totals, or the grand total in a two-way frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A row total counts every interior category in that row. We’ll locate the requested margin, add the two displayed entries without swapping labels, place the sum in the total column, …
Preview problemFind a joint probability from a two-way frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A joint probability uses the single interior cell where both conditions hold and compares it with the grand total. We’ll identify that overlap count, avoid row or column margins in …
Preview problemFind a marginal probability from a two-way frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A marginal probability starts at a row or column edge total and compares it with the grand total. We’ll locate the requested margin, place it over the entire sample, reduce …
Preview problemFind a row conditional probability from a two-way frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A row conditional treats that row as the entire reference group. We’ll use the row total as denominator, use the target cell within it as numerator, reduce and convert the …
Preview problemFind a column conditional probability from a two-way frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A column conditional changes the denominator to the total of the conditioned column. We’ll identify that column margin, use the target row’s cell as the favorable count, reduce the ratio, …
Preview problemInterpret joint, marginal, and conditional probabilities from table counts
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
An intersection with the grand total as denominator is a joint probability, not a conditional one. We’ll interpret both event conditions, locate their shared cell, identify the entire sample as …
Preview problemDecide whether two variables are independent by comparing a marginal probability to a conditional probability
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
A conditional independence test asks whether restricting to B changes A’s rate. We’ll compare the conditional probability directly with A’s marginal probability, measure their difference, and interpret exact equality as …
Preview problemDecide whether two events are independent from marginal and joint probabilities
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
The joint form of the independence test compares the observed intersection with the product of the marginals. We’ll multiply the two given rates, place that expected joint beside the observed …
Preview problemComplete a row-relative-frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
Row-relative frequencies use a different denominator for each row. We’ll divide every cell by its own row total, reduce the first distribution, repeat for the second row, and verify that …
Preview problemComplete a column-relative-frequency table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
Column-relative frequencies normalize each column independently. We’ll divide the upper and lower cells by their own column margin, reduce both columns’ fractions, and use a sum of one within each …
Preview problemDecide whether two variables show an association from a two-way table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
Association is judged by comparing the same outcome’s conditional rate across groups and considering the size of the gap. We’ll align the two bus-use rates, subtract to find their percentage-point …
Preview problemChoose the correct denominator for a percent from a two-way table
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
In a percent-of phrase, the group named after of supplies the denominator. We’ll identify freshmen as the conditioned reference group, locate the bus-riding intersection as the numerator, use the entire …
Preview problemDraw a bounded association conclusion from conditional percentages
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.
Association is evaluated by comparing the same outcome’s conditional rate across groups. We’ll align the two math-preference percentages, calculate and orient the percentage-point gap, state the sample association and higher-rate …
Preview problemIdentify and interpret the parts of a conditional probability statement
Explain conditional probability and independence in everyday language and situations.
A conditional statement describes the target rate within the group named after the bar. We’ll identify treatment as the reference population, recovery as the measured event, convert the decimal to …
Preview problemTranslate an everyday conditional probability statement by identifying the condition and the target event
Explain conditional probability and independence in everyday language and situations.
The among phrase identifies the conditioned group, while the later behavior names the target event. We’ll place sports after the conditional bar, soccer before it, convert the percentage to a …
Preview problemRewrite probability notation as a plain-English claim
Explain conditional probability and independence in everyday language and situations.
Plain-English conditional meaning begins by restricting attention to the event after the bar. We’ll identify B as that reference group, identify A as the outcome counted within it, convert the …
Preview problemRewrite a plain-English probability claim in notation by identifying the condition and the event
Explain conditional probability and independence in everyday language and situations.
The words of students in band make band the denominator group, while playing sports is the measured outcome. We’ll place the target before the bar and condition after it, attach …
Preview problemDecide whether repeated trials in a context are independent
Explain conditional probability and independence in everyday language and situations.
Repeated trials are independent when earlier outcomes do not alter the next trial’s probability distribution. We’ll verify that every die roll retains the same faces and per-face probabilities, explain why …
Preview problemAssess whether association is established, mechanism-based, or data-dependent
Explain conditional probability and independence in everyday language and situations.
Variable names alone cannot establish association without comparable data. We’ll identify the missing within-category pass rates, define how each would be computed, set the comparison rule for similar versus meaningfully …
Preview problemEvaluate a real-world conditional probability claim
Explain conditional probability and independence in everyday language and situations.
Reversing a conditional changes its reference group and generally changes its value. We’ll interpret the supplied disease-conditioned test rate, write the positive-test-conditioned claim separately, compare their denominators, and identify prevalence …
Preview problemVerify unchanged likelihood under independence
Explain conditional probability and independence in everyday language and situations.
The unchanged-likelihood test compares A’s overall probability with A’s probability inside B. We’ll list the fair-die outcomes, compute the even marginal, restrict the sample space to multiples of three, compute …
Preview problem