The two extreme cases, and the difference between one outcome and many

What to assume they know

• Every description of a happening picks out a part of the outcome list — an event is any subset of the sample space, and it occurs when the realised outcome belongs to it
• The empty set, and that it is a subset of every set
• That a set is a subset of itself
• Listing the eight outcomes of three coin tosses in order, without repetition
• Counting the elements of a small set reliably

What they should be able to do

• Produce a description, for a stated experiment, whose subset is empty, and one whose subset is the whole sample space
• Name the two extreme events using the chapter's terms and say which subset each denotes
• Explain why the definition of an event cannot be narrowed to exclude them
• Classify a given event as having one sample point or more than one, and use the chapter's names for the two cases
• Justify that a sample space of n distinct elements carries exactly n one-point events, by pairing each with its element
• Show that the two size-based headings of §14.1.2 — the one-point kind and the more-than-one-point kind — leave the empty event out, and say where the chapter does classify it
• Show that the three headings do not partition the events either, and give the reason: whenever the sample space holds more than one outcome, the sure event answers to the first heading and to the more-than-one-point heading at once
• Read the containments among three descriptions of the same experiment, and say which event is the intersection of the other two

Where it usually goes wrong

• "Simple means easy to describe." It means one sample point and nothing else. The three-coin event described in four words holds three points and is compound; a one-point event may need a long sentence to pin down.
• **"A compound event is two events joined by and or or."** Compound is a statement about size, not about construction. Each of the chapter's three three-coin events comes from a single phrase, and all three of them are compound — the page says so in as many words, and their sizes are three, seven and four.
• "Every event is either simple or compound." The empty event is neither, and the chapter's own numbering invites the error by listing three headings that look like a complete sorting.
• "The impossible event is not an event, it is the absence of one." It is a subset, so the definition takes it, and the chapter reaches it from a description about a die rather than by fiat.
• "The sure event carries no information." It is the event whose probability the axioms will fix outright, which is what makes every other probability measurable against something.
• "An outcome and the one-point event holding it are the same object." One belongs to the sample space, the other is a subset of it. The chapter's own Note on p. 296 concedes it will write them alike for convenience — a convenience, not an identity.
• "There are as many events as outcomes." There are as many one-point events as outcomes. Every other subset is an event too, and they far outnumber them.
• "At most one head and at least one head are opposites." They overlap, in exactly the outcomes with one head, and their union is everything. The pair that really are opposites is at-least-one against none-at-all.

Questions to check understanding

• Given an experiment and a description, decide whether the event is empty, the whole sample space, or neither
• Count the one-point events of a stated sample space, and justify the count
• Classify listed events by size — Exercise 14.1 Q4 does exactly this for three tossed coins, asking which of four named events are one-point and which hold more, alongside a mutual-exclusivity question
• Supply a description that produces the empty event for a stated experiment
• Exercise 14.1 Q2 includes a description of a die throw that no face satisfies and one that every face satisfies, so a single question drills both extremes
• Given two descriptions, state the containment between their events

Examples worth working on the board

Values marked verified are an added counting against the printed page.

• The die, and a face that is a multiple of seven (§14.1.2, p. 290). One throw, so S = {1, 2, 3, 4, 5, 6}. Verified: the smallest positive multiple of 7 is 7 itself, which exceeds every face, so the subset is empty. The chapter poses this as a question to the reader before answering it.
• The die, and a number that is odd or else even (§14.1.2, p. 290). Verified: each of the six faces is one or the other, so the subset is the whole of S and nothing is left out. Two descriptions, the same die, opposite extremes.
• Two coins, and every one-point event (§14.1.2, p. 290). S = {HH, HT, TH, TT}. The chapter names all four one-point events explicitly and labels them with subscripts 1 to 4. Verified: four elements, four such events, and the pairing is forced — each is fixed by naming its single member.
• Three coins, and three compound events (§14.1.2, p. 291). S has eight outcomes.
  • exactly one head → {HTT, THT, TTH}
  • at least one head → {HTT, THT, TTH, HHT, HTH, THH, HHH}
  • at most one head → {TTT, THT, HTT, TTH}

Verified: the sizes are 3, 7 and 4. The seven-element one is everything but TTT, which is the only outcome with no head at all. The four-element one is the three single-head outcomes together with TTT.

• How those three fit together. Verified, and none of this is printed: the three-element event sits inside both of the others; the intersection of the at-least-one and the at-most-one events is exactly the exactly-one event; and their union is the whole sample space, because any outcome either has a head or has none. Deriving that overlap is the cleanest possible preview of the algebra of events, and it uses no vocabulary the chapter has not yet given.
• The counting behind "exactly n". Verified: pairing each one-point event with the element it holds is a match with no leftovers on either side, which is why the count of one-point events equals the count of outcomes and cannot drift from it.
• How rare one-point events are. Verified, and not printed in this chapter: a four-outcome list carries 16 events in all and 4 of them hold one point; an eight-outcome list carries 256 events and 8 of them do. The chapter counts its one-point events and never counts its events, so the totals must be presented as arithmetic added here.
• What the size headings leave out, and where the chapter puts it anyway. Verified: the empty event holds no sample point, so it is neither the one-point kind nor the more-than-one kind — the last two headings do not reach it. It is not unclassified, though: the first heading names it outright. §14.1.2 prints the three headings and remarks on neither the omission nor the fact that the first heading closes it.

Figures to have open

• One box labelled S carrying a shaded region, drawn twice: shading nothing, then shading everything. Standard schematic; the chapter prints no picture in §14.1.2, verified against the page images of pp. 290–291.
• An eight-cell strip for the three-coin outcome list, reused across sections 8 and 9 so the three highlights can be overlaid. An added device.
• A one-to-one pairing diagram, outcomes on the left and one-point events on the right, four of each. Standard schematic.
• No figure from the textbook is needed. The chapter's only numbered figure, Fig 14.1 on p. 301, belongs to the addition-rule topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 14 "Probability", §14.1.2 "Types of events", pp. 290–291, comprising the three numbered headings for the impossible and sure events, the one-point event and the compound event
• The Note in §14.2, p. 296, which prints the alternative name for the one-point event whole and settles the notation for it
• Exercise 14.1 Q2 and Q4, pp. 294–295
• The Summary, p. 312, whose second and third bullets restate the two extremes