Independence, and why it is a different idea from having no outcome in common

What to assume they know

• The definition of a conditional probability and its side condition
• The product form from §13.3, since one definition here is that form
• Events as subsets of a sample space, and the empty intersection
• Complement of an event, and that an event and its complement partition the space
• The addition rule for a union, from Class XI
• Comfort with checking whether two given numbers multiply to a third

What they should be able to do

• Compute both conditionals for a suit-and-rank pair on a deck and observe that each equals the corresponding unconditional value
• State the chapter's first definition of independence, with both of its nonzero conditions
• Derive the product form from the first definition and state it as the second definition
• Say what the product form can do that the conditional form cannot, and why the chapter presents both
• Define dependence as the failure of the product equality
• Distinguish independence from mutual exclusion by saying what kind of object each is a statement about
• Prove that two events with no shared outcome and nonzero probabilities cannot be independent
• Test a given pair for independence by computing three probabilities and comparing
• Prove that independence survives replacing either or both events by their complements
• Express the chance of at least one of two independent events using only their complements
• State the four conditions for three events, and produce a case satisfying the first three but not the fourth

Where it usually goes wrong

• "Mutually exclusive events are independent — they have nothing to do with each other." Exactly backwards. Learning that one of them happened tells you the other certainly did not, which is the strongest dependence there is. This is the misconception the section is built around; give it a full section, not a line.
• "Independent means unrelated in the real world." It means three numbers satisfy an equation. Example 12 has three events on one sample space, all describing the same three coins, and one pair of them is independent while two are not.
• "If they overlap they are not independent." Independent events usually do overlap. The ace of spades is in both events of the opening argument.
• "You only need to check one conditional." In principle either will do, but each needs its own denominator to be nonzero. The product form avoids the question, which is why the chapter reaches for it in every example.
• "The two definitions are the same statement." They agree whenever both probabilities are nonzero, and only the product form says anything when one is nought. Do not narrate them as interchangeable.
• "Three events are independent if the three pairs are." They are not. The fourth condition is separate and can fail on its own; the two-coin case in Worked examples shows it happening.
• "If two events are independent, replacing one by its complement breaks it." It does not. Example 13 proves one case and the Note states two more.
• "An exercise printed under this heading is about independence." Not reliably. Exercise 13.2 also carries the multiplication-rule practice for §13.3, which has no exercise of its own, and Q16 is a plain conditional-probability question. A student who assumes the heading applies to every item will look for independence in questions that do not contain any.

Questions to check understanding

• Given three probabilities, decide whether a pair is independent, and say which comparison settles it
• State both definitions of independence and say which one needs a side condition
• Prove that two events with no shared outcome and nonzero probabilities are not independent
• Given a union probability and one event's probability, solve for the other twice — once assuming exclusion and once assuming independence — in the shape of Exercise 13.2 Q7
• Test three named events on a small sample space and classify each of the three pairs, in the shape of Example 12
• Prove that independence survives complementing one of the two events
• Compute the chance of at least one of two independent events using complements
• Produce three events that are pairwise independent but not independent together

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The opening deck argument (§13.4, Part II p. 417). One card, two events: a spade, and an ace. Verified: thirteen over fifty-two is a quarter, four over fifty-two is one thirteenth, and the single card that is both gives one over fifty-two. Both conditionals are then computed and both come out at the corresponding plain probability. The chapter runs the calculation in both directions on purpose — a student who sees only one direction will think independence is a one-way property.
• Definition 2 (Part II p. 418), the conditional form: each conditional equals the corresponding plain probability, and the definition carries both nonzero conditions explicitly.
• Definition 3 (Part II p. 418), the product form, obtained by feeding Definition 2 into the multiplication rule. What the chapter does not say, and what: the product form needs no side condition at all, because nothing is divided. It therefore applies to events the first definition cannot reach. See Notes — this is an added observation and it is the sharpest mathematical point in the topic.
• Remark (i) (Part II p. 418). Dependence is defined as the failure of the product equality, and nothing more. Note it is defined off the product form, not off the conditional form, which is consistent with the point above.
• Remark (ii) (Part II p. 418), the longest remark in the chapter and the reason this topic exists. It makes three separate claims: that one term is about probabilities while the other is about sets of outcomes; that two mutually exclusive events share nothing while two independent events may share plenty; and that with nonzero probabilities neither condition can hold alongside the other. The chapter asserts the third claim and does not prove it. Verified, and the proof is one line: events with no shared outcome have intersection probability nought, while the product of two nonzero probabilities is not nought, so the product equality fails. Show that line; it turns a warning into a theorem.
• Remark (iii) (Part II p. 418), independent experiments. Distinct from independent events and easy to skip. It is the licence behind every question that tosses a coin and throws a die in the same breath, including Exercise 13.2 Q4.
• Remark (iv) (Part II pp. 418–419), three events. Four conditions: the three pairwise products and the triple product. The chapter then says that failing any one of them costs independence. It gives no case where the three pairwise conditions hold and the fourth fails, which is the whole reason the fourth condition is listed separately. Verified, and this is an added example, built from apparatus the chapter uses constantly: toss two fair coins; let the first event be a head on the first coin, the second a head on the second, and the third that the two coins agree. Each has probability one half; each pair meets at the two-head outcome or at the two-tail outcome and so has intersection probability one quarter, which is the product; but all three together happen only on two heads, giving one quarter where the triple product demands one eighth. Label it clearly as not printed in the book.
• Example 10 (Part II p. 419), one die. Verified: the multiples of three give one third, the even faces one half, and the single face six gives one sixth, which is the product. Independent.
• Example 11 (Part II p. 419), a die thrown twice, odd on each throw. Verified: one half, one half, and one quarter. Independent — and this is the case where independence is obvious from the physics before any arithmetic, which makes it the right one to use for the point that the definition is a check, not an intuition.
• Example 12 (Part II pp. 419–420), three coins and three events: all alike, at least two heads, at most two heads. Verified from the chapter's own outcome lists: probabilities one quarter, one half and seven eighths; the first pair meets in one outcome giving one eighth, which equals the product, so independent; the first and third meet in one outcome giving one eighth against a product of seven thirty-seconds, so dependent; the second and third meet in three outcomes giving three eighths against a product of seven sixteenths, so dependent. Three verdicts from one sample space is the best single exhibit in the section.
• Example 13 and Fig 13.3 (Part II p. 420). Independence survives replacing the second event by its complement. The proof splits the first event into the part inside the second and the part outside, subtracts, and factorises. Read off the figure: the rectangle is the sample space, the two circles are the two events, and four labels with arrows name the four regions — outside both, first only, both, and second only. The two crescents are shaded and the lens where the circles overlap is left white, which is not what the proof is about; the proof is about the first circle being cut in two. If the figure is shown, redraw it with the first circle shaded whole and the cut drawn across it.
• The Note (Part II p. 421). Two more consequences stated without proof: complementing the first event alone, and complementing both. Verified: both follow by running Example 13's argument a second time, the second case by applying the first result to the outcome of the second.
• Example 14 (Part II p. 421). At least one of two independent events, rewritten as one minus the product of the two complements. Seven printed lines, of which the interesting one is where the addition rule's subtracted term is replaced by the product. Verified by the alternative route, which is shorter and worth showing beside it: at least one fails only when both fail, and by the previous result the two failures are independent.
• Exercise 13.2 (Part II pp. 421–423), eighteen items, of which these carry the topic: Q1, a bare product — verified three twenty-fifths; Q4, a coin and a die, which is Remark (iii) in action — verified independent, one half times one sixth is one twelfth; Q5, a die whose faces are coloured, where the two events overlap in one face — verified not independent, one sixth against one quarter; Q6, three given numbers to test — verified not independent, one fifth against nine fiftieths; Q7, which solves for the unknown probability twice, once under each hypothesis — verified one tenth if the events cannot both happen, one fifth if they are independent, and the two answers being different is the cleanest possible demonstration that the two conditions are not the same condition; Q8, four parts on a stated independent pair — verified nought point one two, nought point five eight, nought point three and nought point four, the last two being the two definitions read straight back; Q9 — verified three eighths; Q11 — verified nought point one eight, nought point one two, nought point seven two, nought point two eight; Q12 — verified seven eighths, by complementing; Q14 — verified two thirds and one half; Q15, three suit-and-rank pairs — verified independent, independent, and not independent; Q16, a hostel readership question that is a conditional-probability item wearing an independence exercise's clothes — verified one fifth, one third, one half; Q17 — verified one over thirty-six; Q18 — verified the option giving the product of the two complements.
• Exercise 13.2 Q10 is unsatisfiable as printed (Part II p. 422). See Notes. Do not set it.

Figures to have open

• Two overlapping circles inside a rectangle for section 7, drawn twice at the same size: once with the overlap emptied to show mutual exclusion, once with it full to show independence with a shared region. This is an added redrawing; the chapter's Fig 13.3 on Part II p. 420 belongs to a different argument and shades the wrong regions for this one.
• A redrawn version of Fig 13.3 for section 10 with the first circle shaded whole and a single cut across it, so the proof's split is what the eye sees.
• A three-row table for section 9 built with the repo's DataTable component. All three rows must share one fitSize; the three-coin row has the longest entries and will otherwise typeset smaller and read as a footnote to the other two.
• A two-coin outcome square for section 12 with the three events drawn as three overlays on the same four cells, so the student can see all three pairwise intersections and the triple intersection at once. Not in the book.
• No figure is needed for sections 4 to 6. They are notation, and Notation is the right carrier.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 13 "Probability", §13.4 Independent Events, Part II pp. 417–421
• The opening deck argument, Part II p. 417
• Definition 2 and Definition 3, and Remarks (i) to (iv), Part II pp. 418–419
• Examples 10, 11 and 12, Part II pp. 419–420
• Example 13 with Fig 13.3, Part II p. 420
• The Note on complements, and Example 14, Part II p. 421
• Exercise 13.2, Part II pp. 421–423
• Summary, fourth bullet, Part II p. 437