How can I tell whether two probability events are independent?
Use conditional probability or a product check to test independence, then compare how the answer changes under different experiment rules.
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How does drawing with replacement differ from drawing without replacement?
See how putting an item back changes the next-draw probabilities, with side-by-side trees and a worked marble example.
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How can a probability tree help me keep multi-step cases straight?
Build a labeled tree for sequential events, follow one path at a time, and combine the paths that match your question.
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Why doesn’t a random sequence have to look evenly mixed?
See why streaks can appear in random sequences, compare two examples, and avoid reading a small sample as a prediction.
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How can I calculate the chance that either of two events happens without double-counting?
Use the addition rule to count outcomes in either event once, subtracting the overlap when the events share outcomes.
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How can I calculate the chance of two events both happening?
Choose the right probability rule by checking whether the first event changes the second, then calculate and explain the result.
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How does the complement rule help calculate the chance of at least one event?
Learn when to subtract the chance of none from 1, work through a fair-die example, and use a template to set up your own calculation.
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How can I tell whether outcomes in a probability problem are equally likely?
Use a quick assumption check to decide whether counting outcomes is valid, and see what to do when equal chances or probability information are missing.
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How can I make a game’s setup repeatable without making every play identical?
Separate setup rules players need every time from a few controlled choices, then record and test those choices without changing everything at once.
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Why does a probability calculation need a clearly defined set of outcomes?
See how counting two die rolls as ordered pairs or sums changes the counting method—and why the sums cannot simply be treated as equally likely.
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How do I design a short version of a game that preserves its central decision?
Identify the game’s defining choice, shorten repeated steps, and check that the choice still has meaningful consequences.
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How do I make a resource choice involve a real trade-off?
Use a resource trade-off table to compare what each choice costs, what it enables, and how its consequences affect later decisions.
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How can a small set diagram clarify an argument about groups?
Translate claims about groups into circle relationships, then check whether the diagram supports the argument’s conclusion.
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Why can’t I conclude that two groups overlap just because they share a category?
Learn why two groups can both belong to a larger category without sharing members, and use a simple diagram and inference check.
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How can I check a claim that says “some” without assuming it means “most”?
A comparison sheet for reading “some” carefully, using one example correctly, and avoiding unsupported conclusions about a group.
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How do I distinguish a contradiction from two claims that merely seem different?
Align the subject, conditions, and meanings of two claims, then check whether one is the exact negation of the other.
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What is the difference between “not both” and “neither”?
Learn why “not both” still allows one item to be true, while “neither” means both items are false, using all four cases.
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How can I check a conditional rule by listing its possible cases?
Use four rows to test every truth-value combination and identify the one case that makes a conditional false.
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How can I tell whether two conditions are both required or either one is enough?
Use the rule’s wording and a four-case decision table to tell whether it requires both conditions or accepts either one.
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How can I compare a table and graph that summarize the same small dataset?
Use the table to check exact values and the graph to see the pattern, then verify that labels, units, and plotted values match.
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How can I tell whether a displayed average is weighted or unweighted?
Learn how group sizes affect a combined average, work through a small example, and ask for the figures needed to check the method.
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How do subtotals help me understand a larger table?
Learn how a subtotal groups component rows, how it fits into a grand total, and how to avoid counting the same entries twice.
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How should I read a table that counts the same items in different categories?
Learn to tell whether category counts describe separate groups or overlapping labels, with a worked example and a checklist for checking totals.
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How can a cumulative total be different from a value for one period?
See how period-by-period values differ from a running total, and learn why a cumulative line can rise while the latest period’s value falls.
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