Read probability examples in Book Isle: separate mutually exclusive and independent events with four outcomes

When reading probability, memorising two definitions can leave mutual exclusion and independence confused. Use Book Isle to locate the text and retain bookmarks and notes, then list four outcomes on paper or in your own notes and check them individually. The example assumes two fair, mutually independent coin tosses; it neither requires real tossing nor claims that four trials must produce one of each outcome. Explain why one event pair is independent and another mutually exclusive, rather than expecting the app to solve probabilities.

Record assumptions and the source location

Choose an accessible introductory probability example and record title, author, edition, chapter, subsection and a distinctive nearby phrase. Keep fixed page numbers where useful, but do not rely only on screen page counts in reflowable EPUB. Bookmark or briefly annotate the passage with three assumptions: fairness, independence of the two tosses and ordered recording. If the book omits one, label it as an additional assumption for this exercise rather than attributing it to the author. Read the question before the answer and retain uncertainties as unconfirmed. Search locates text but does not determine implicit conditions. Do not yet equate the everyday phrase unrelated with probabilistic independence.

List four ordered basic outcomes

Write HH, HT, TH and TT in your own record, using H for heads and T for tails. The first letter represents toss one and the second toss two. A two-by-two layout can use rows for the first toss and columns for the second. Do not merge HT with TH: their order differs. Each pair is one complete two-toss outcome, not two separate list entries. Under the fair-and-independent assumptions, each cell has probability 1/4 and the four sum to 1. Check cell count and total first. A list of two heads, one of each and two tails must expand its middle category into two ordered outcomes rather than assigning 1/3 to each category.

Express events as sets of cells first

Let A mean H on the first toss, containing HH and HT, and B mean H on the second, containing HH and TH. Record each event’s full meaning in Chinese or English before its symbol rather than leaving A and B to be guessed later. Mark their cells in two ways. Only HH is shared, so P(A)=1/2, P(B)=1/2 and P(A∩B)=1/4. The intersection means one complete two-toss outcome satisfies both conditions, not that another experiment is performed. If marked cells and formulas disagree, inspect letter positions against the event definitions rather than directly changing probability values to fit an answer. Explain why every listed outcome belongs to its event.

Check independence by product and conditioning

Compute P(A)P(B)=(1/2)×(1/2)=1/4 and compare it with the intersection, confirming independence here. Then restrict attention to B’s two cells, HH and TH. A occurs only in HH, so P(A|B)=1/2, the same as P(A). The vertical bar means given that B occurred, not division of the letters A and B. The two checks should agree, but conditional probability requires a nonzero denominator. Do not carry the result over if the book changes events or assumptions. Independent tosses do not make every pair of events built from them independent; examine a different pair next.

Repeat the checks for mutually exclusive events

Keep A as H on the first toss and define C as T on the first, containing TH and TT. A and C share no cell, so their intersection probability is 0 and they are mutually exclusive. But P(A)P(C)=1/4 rather than 0, so they are not independent. Given C, A is impossible: P(A|C)=0 instead of its original 1/2. Record two separate questions: can both occur, and does knowing one occurred change the other’s probability? Do not write the false slogan mutually exclusive therefore independent. Both events here have positive probability; retain that condition when generalising instead of mixing zero-probability cases into the same statement.

Match textbook terminology to your own grid

Return to the passage in Book Isle and search for the relevant term, checking whether the author’s events actually match A/B or A/C. If the author uses at least one H, mark HH, HT and TH afresh rather than applying the first-toss event merely because H appears. Record your event wording separately from the book’s example number and retain a short locating phrase. If copying a formula corrupts the intersection symbol, verify it on the original page; writing both conditions hold is better than keeping the wrong symbol. Separate the book’s statement from your verification so invented events are not presented as the original exercise.

Explain without the answer and diagnose errors

Hide or leave the answer and explain from your four cells alone why A and B can occur together and are independent, while A and C cannot occur together and are not independent. If you can only recite the product formula, identify the intersection cells as well. If you obtain 1/4 for the conditional probability, check whether you forgot to restrict its denominator to B’s two cells. Do not replace this assumed-model argument with frequencies from a few real tosses; a short sample’s variation does not change the grid definitions. Confirm consistent H/T meanings throughout. Completion means reconstructing calculations from sets, not recognising an answer after seeing it.

Keep checkable assumptions and corrections

Retain model assumptions, four outcomes, three event definitions, intersections, both pair judgements and the book location. If later reading introduces an unfair coin, mark a new model rather than replacing 1/2 while still calling the four cells equally likely. Add a correction reason beside an incorrect judgement, such as confusing mutual exclusion with independence, instead of silently rewriting history. Save a note copy through the export options available in your installed Book Isle version and reopen it to check formulas and symbols. For the next exercise, change only one event and rebuild the marked cells, preserving the sequence from wording to sets to numbers.

Related tutorials and sources

Follow worked examples in Book Isle: track symbols, units and substitutions; Book Isle question-led chapter reading: separate initial questions from final answers.

Sources checked on 5 November 2026: the official Book Isle App Store listing and Apple Lookup show version 1.1.3, requiring iOS / iPadOS 15.6 or later. This workflow uses local reading, search, bookmarks and notes for manual checking. Readers record the samples and conclusions; automatic problem solving, experiment analysis or chart recognition is not assumed. Confirm available controls against the installed version and file type. The exercise does not claim step-by-step device testing in this batch.

Method reference: OpenStax, Introductory Statistics 2e: Independent and Mutually Exclusive Events. The grid record, Book Isle workflow and event comparisons are original exercises; their probabilities apply only under the stated assumptions.