The table gives the numbers of people with or without feature A (A′) and with or without feature B (B′). One person is chosen at random. Find P(A | B).
B | B'; A | 12 | 6; A' | 14 | 22
Answer: 6/13
10th Grade Conditional Probability and the Rules of
Imaginesis: printable 10th Grade Conditional Probability and the Rules of Probability worksheets for the United States (Common Core) — Using Total Probability and Bayes' Formula, Finding Conditional Probability from a Table, Finding the Probability of a Union of Events: word problems with hints, answer keys and step-by-step solutions as PDF, also as a test in groups A–D.
Sample word problems
The probabilities of events A and B are given. Find the probability of the union A ∪ B.
P(A) = 12/30, P(B) = 14/30, P(A ∩ B) = 3/30
Answer: 23/30
Skills in this topic
Using Total Probability and Bayes' Formula
Students learn to calculate overall event chances across multiple stages and find reversed conditional probabilities using two-urn models.
HSS-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
At this level, students apply the general multiplication rule to find the probability of combined events. Working with multi-step experiments, they calculate the total probability of an outcome that can happen through different paths, and use Bayes' formula to determine conditional probabilities in reverse—figuring out which initial path was taken after observing the final outcome.
A common mistake is confusing the condition with the outcome, treating the probability of drawing a specific item from an urn, P(ball | urn), as identical to the probability that a specific urn was picked, P(urn | ball). Students also frequently forget to multiply by the probability of selecting the first container itself, focusing only on the ratio of colored balls inside it.
Practice tasks center on two-urn experiments. For example, a student is given two urns containing different mixtures of colored marbles, such as red and green. In a total probability task, they determine the overall likelihood of drawing a red marble after randomly picking an urn. In a Bayes' formula task, they are told that a red marble was drawn and must calculate the probability that it came from the first urn.
Finding Conditional Probability from a Table
Students learn to find the probability of an event given another condition using data from a table, expressing the answer as a fraction and explaining its meaning.
HSS-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.HSS-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.HSS-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.HSS-CP.B.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Students learn to calculate the conditional probability of an event A given that another event B has occurred. They use data to identify the subset of outcomes that satisfy condition B, make that count their denominator, and find the fraction of those specific outcomes that also belong to A. Students then interpret what this fraction represents within the context of the problem.
A frequent error is dividing by the total population rather than the restricted condition. When asked for the probability of A given B, students often pull the grand total from the entire data set instead of restricting their focus to the row or column representing only condition B.
Worksheet tasks feature two-way frequency tables categorizing survey results or experimental data. Students read the table to locate the given condition, identify the overlapping category, and write the resulting conditional probability as a simplified fraction.
Finding the Probability of a Union of Events
Students learn to use the Addition Rule to calculate the probability that at least one of two events occurs, accounting for any outcomes they share.
HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
At this level, students calculate the probability that event A, event B, or both occur within a sample space. They apply the Addition Rule, represented as P(A or B) = P(A) + P(B) – P(A and B), interpreting outcomes described with "or," "and," and "not." Students work with probabilities expressed as fractions, decimals, or percentages to model real-world situations.
The most common error is forgetting to subtract the intersection, P(A and B). When two events can happen at the same time, simply adding their individual probabilities double-counts the shared outcomes, which frequently results in an inaccurate answer or a probability greater than 1.
Tasks based on the Probability of a union of events template typically give students a scenario—such as drawing a card with specific characteristics or selecting a person from survey categories. Students identify the probability of each separate event, determine the probability of the overlap where both conditions are met, and apply the formula to find the combined probability.