Events that cannot both happen, and events that between them must

What to assume they know

• "Or", "and" and "not" are the three set operations under new names — union, intersection, difference and complement read as descriptions of events
• The two extreme cases, and the difference between one outcome and many — the extreme events, and events of one sample point
• Disjoint sets and the empty set, from the chapter on sets earlier in this book
• Writing the 36 outcomes of two thrown dice as ordered pairs, and computing the total on each
• Reading indexed notation for a family of n sets and its union

What they should be able to do

• Test a stated pair of events for a shared outcome, and conclude on that basis alone whether they can occur together
• Test a stated family for whether its union is the whole sample space
• Give an example of a family satisfying one condition and failing the other, in each direction
• State the general condition for n events to cover the sample space, and the extra condition that makes the family a clean split
• Explain why one-point events can never occur together, without checking cases
• Use a size count as a check on a family that covers, and say why the check catches an overlap
• Given several described events on one experiment, find every pair that cannot occur together

Where it usually goes wrong

• "Events that cannot both happen must between them cover everything." The refutation is easy to build: two different one-point events on a six-outcome die exclude each other and leave four outcomes unclaimed. Do not reach for §14.1.4's second pair as evidence here — that pair shares outcomes, so it is not an exclusive pair at all, and a non-exclusive family that fails to cover tests neither implication. It is useful for something else: it fails both conditions at once, which is what makes it a poor witness for either.
• "Covering the sample space means the pieces do not overlap." §14.1.5's three-event family covers and overlaps in the outcome 3. This is the chapter's deliberate choice of example and the single most important thing in the section.
• "They almost exclude — they only share one outcome." The condition is not a matter of degree. One shared outcome is a complete failure of it, which is why the chapter can settle the question by producing a single witness.
• "Check one pair and the family is settled." The condition is required of every pair. Example 2 has six pairs to test and five of them fail.
• "A family that splits the sample space cleanly cannot contain a one-point event." Example 3's first member holds exactly one outcome. Nothing forbids it, and the one-point events of a sample space taken all together are themselves such a family.
• "If the sizes add to the size of the sample space, the family is a clean split." Necessary, not sufficient — an overlap and a gap can cancel each other in the total. The check earns its keep only alongside one of the two conditions, and then it settles the other.
• "The odds and evens on a die are the model to copy." They satisfy both conditions simultaneously, which hides the distinction the sections are drawing. Teach it alongside the overlapping example, never alone.
• "Mutual exclusivity is about the descriptions sounding incompatible." It is about the sets. A total below 4 and a total above 11 sound incompatible and are; an odd face and a face below 4 sound incompatible to some students and are not.

Questions to check understanding

• Given several described events on one experiment, name every pair that cannot occur together — the form of Example 2 and of Exercise 14.1 Q3
• Exercise 14.1 Q1 asks the question at its smallest, for a die showing 4 against a die showing an even face, where one event sits wholly inside the other
• Exercise 14.1 Q5 runs the whole grid of possibilities on three tossed coins, asking in turn for a pair that excludes, a triple that excludes and covers, a pair that does not exclude, a pair that excludes without covering, and a triple that excludes without covering
• Exercise 14.1 Q7 sets six true-or-false claims about the two conditions on the two-dice events of Q6 and demands a reason for each
• Given a family, decide whether it covers, and justify by naming an unclaimed outcome if it does not
• Add the sizes of a covering family and account for any excess

Examples worth working on the board

Values marked verified are worked out here against the printed page; the chapter prints the sets and almost none of the counts.

• One die, the odds against the evens (§14.1.4, p. 292). S = {1, 2, 3, 4, 5, 6}, with A described by an odd face and B by an even face, so A = {1, 3, 5} and B = {2, 4, 6}. Verified: the intersection is empty, so no single throw can satisfy both. Verified, and worth flagging when explaining it: this pair also covers the sample space, so it happens to satisfy both conditions at once — a poor example to generalise from, and the chapter does not warn about that.
• One die, the odds against the low numbers (§14.1.4, p. 293). A = {1, 3, 5} as before, and B is described by a face below 4, giving B = {1, 2, 3}. The chapter observes that 3 lies in both and concludes the pair fails. Verified, and not printed: 1 lies in both as well, so the intersection is {1, 3} and holds two outcomes rather than one. The chapter needs only a single witness to reach its conclusion, but an explanation listing the intersection should list both. Verified: the union here is {1, 2, 3, 5}, which omits 4 and 6, so this pair also fails to cover — a single pair failing both conditions.
• The Remark on one-point events (§14.1.4, p. 293). Verified: two distinct one-point events hold different single outcomes, so nothing can lie in both, and the conclusion follows from distinctness alone rather than from any property of the experiment.
• The chapter's covering family (§14.1.5, p. 293). One die again, with three described events: a face below 4, a face above 2 but below 5, and a face above 4. These give {1, 2, 3}, {3, 4} and {5, 6}. The chapter checks that the three unions run out to the whole sample space. Verified, and none of it printed: the first two share the outcome 3, so the family covers without being a clean split; and the sizes are 3, 2 and 2, adding to 7 against a sample space of 6 — the excess of exactly one is the single outcome that got counted twice. Running that arithmetic is the cheapest possible demonstration that covering does not imply non-overlapping.
• Example 2 — two dice and four descriptions (§14.1.5, pp. 293–294). All 36 ordered pairs. The four events are described by an even total, a total that is a multiple of 3, a total below 4, and a total above 11. Verified: the four sets hold 18, 12, 3 and 1 outcomes. The chapter prints all four lists and then tests six pairs. Verified: the even-and-multiple-of-3 overlap is the six pairs totalling 6 or 12; the even-and-below-4 overlap is the single pair of ones; the even-and-above-11 overlap and the multiple-of-3-and-above-11 overlap are both the single pair of sixes; and the multiple-of-3-and-below-4 overlap is the two pairs totalling 3. Only the below-4 and above-11 events have nothing in common, because no total can be under 4 and over 11 at once. Exactly one of the six pairs excludes.
• Example 3 — three coins split three ways (§14.1.5, p. 294). S holds 8 outcomes. The three events are described by no head, exactly one head, and at least two heads, giving {TTT}, {HTT, THT, TTH} and {HHT, HTH, THH, HHH}. Verified: sizes 1, 3 and 4, adding to 8, with every pair sharing nothing and the union running out to the whole sample space. Both conditions hold. Note: the first member holds a single outcome. A family that splits the sample space cleanly may perfectly well contain one-point events, and this one does.

Figures to have open

• A six-cell die strip that can take two shadings at once, so an overlap is visible as a doubly shaded cell. An added device; §14.1.4 and §14.1.5 print no picture at all, verified against the page images of pp. 292–294.
• A six-by-six grid of the 36 ordered pairs with each cell carrying its total. This is the single most useful asset in the chapter and is reused by later topics. Not in the book; the chapter lists the pairs in running text.
• A block diagram of the sample space cut into three, drawn once with a seam overlapping and once with the seams clean, for section 10's contrast.
• No textbook figure is required. Fig 14.1 on p. 301 is the chapter's only numbered figure and belongs to the addition-rule topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 14 "Probability", §14.1.4 "Mutually exclusive events", pp. 292–293, including the Remark on p. 293
• §14.1.5 "Exhaustive events", pp. 293–294, including the general statement for n events and Examples 2 and 3
• Exercise 14.1, pp. 294–295, questions 1, 3, 5 and 7
• The Summary, p. 312, whose eighth and ninth bullets restate both conditions
• Forward pointer: axiom (iii) of §14.2, p. 296, is stated for a pair satisfying the exclusion condition, which is why these sections come first