Use sample spaces, complements and unions without double counting.
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Try
Roll a fair six-sided die. Define A as an even result and B as a result greater than four. Before using a formula, list A as2,4,6 and B as5,6. Their overlap contains6. Counting each list in full counts6 twice, explaining why a union calculation needs a subtraction.
Predict
Predict P(A or B) and compare it with P(A)+P(B). Also predict whether disjoint events with positive probabilities can be independent. If learning that A occurred rules out B, it changes the probability of B, which already suggests the answer.
Experiment
The union is2,4,5,6, so its probability is4/6=2/3 under the fair-die assumption. The formula gives3/6+2/6−1/6=4/6. Draw two overlapping circles or mark the six outcomes on a list; both representations should count every union outcome exactly once.
For three independent fair coin flips, compute the probability of at least one head through its complement. All tails has probability(1/2)³=1/8, so at least one head has probability7/8. Compare this with listing the seven successful sequences. The complement avoids extra counting while using exactly the same model.
Understand
A sample space contains the possible outcomes for a specified random process. An event is a set of outcomes. Probabilities are nonnegative, the whole sample space has probability one, and disjoint alternatives add. Counting favorable outcomes divided by total outcomes is valid only when those outcomes are equally likely. A list alone does not make them so.
The complement Aᶜ means A does not occur and has probability1−P(A). The union A or B includes outcomes in either event, including both. The intersection A and B includes only common outcomes. Inclusion–exclusion states P(A or B)=P(A)+P(B)−P(A and B). State whether ordinary-language “or” is inclusive; in this unit it is unless explicitly qualified.
Disjoint events cannot both occur. Independent events do not change each other’s probabilities under conditioning, equivalently their joint probability equals the product. If disjoint events both have positive probability, the intersection is zero but the product is positive, so they are not independent. These terms describe different structures.
Probability assignments must be mutually coherent. Proposed values P(A)=0.7, P(B)=0.6 and P(A and B)=0.1 imply a union of1.2, which is impossible. The overlap must be at least0.3 and cannot exceed either marginal probability. Checking such bounds can uncover a mistaken denominator before further calculation.
An observed relative frequency estimates a model probability but is not identical to it. Thirty sixes in one hundred rolls neither proves a true probability of0.30 nor establishes fairness. Random variation, selection and the physical mechanism all matter. Repeated trials under comparable conditions provide evidence, interpreted through an explicit model rather than a guarantee of exact frequency matching.
Apply
Write two descriptions of the same event: one directly and one as a complement. Identify where independence enters your calculation rather than multiplying merely because two numbers are available. For an unfamiliar context, such as simultaneous service failures, explain why a shared power supply could invalidate an independence assumption.
Continue
Next condition on observed information. The numerator may remain the same while the relevant denominator changes.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
P(A)=2/5, P(B)=1/2 and P(A and B)=1/5. Find P(A or B).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Three independent fair coin flips are made. Find the probability of at least one head.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For a fair six-sided die, find the probability of a value that is even or greater than4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Two disjoint events each have positive probability. They are:
Can P(A)=0.7, P(B)=0.6 and P(A and B)=0.1 describe one probability model? Explain.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A die shows six on30 of100 rolls. Explain why the observed proportion does not prove either that the die is fair or that P(six)=0.30.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
