14. Probability

8 topics2 h

Chapter 14 of NCERT Mathematics for Class 11: Probability. Three sections — Events are subsets, The algebra of events and Probability laid on axioms. Eight videos, 2 h in all.

Worked answers to the 38 exercise questions in this chapter: Exercise 14.1 · Exercise 14.2 · Miscellaneous Exercise

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Part 1

Events are subsets

16 minEvery description of a happening picks out a part of the outcome list'At least one tail' and 'at most one head', asked of two tossed coins, name different faces of the coin - and pick out the identical three outcomes.16 min · 17 parts18 minThe two extreme cases, and the difference between one outcome and manyAsk a die for a face that is a multiple of seven, and nothing answers - an ordinary sentence naming a collection that happens to hold nothing at all.18 min · 17 parts

Part 2

The algebra of events

17 min"Or", "and" and "not" are the three set operations under new names'Or' has to mean one or the other or both, or the trade between English sentences and sets collapses. Read it exclusively instead, and two readings of one sentence disagree on 781 of 1024 tested pairs.17 min · 17 parts18 minEvents that cannot both happen, and events that between them mustEvents that cannot both happen, and events that between them leave nothing out, sound like one condition stated twice. A die's odd faces and low faces overlap, and still miss two.18 min · 17 parts

Part 3

Probability laid on axioms

24 minThree conditions on a function, in place of a recipe for countingA coin admits infinitely many legitimate probability assignments, one for every number between 0 and 1. The axioms never say which is correct for a real coin.24 min · 17 parts17 minAssuming the outcomes are equally likely recovers the counting ruleToss a fair coin twice, score a point for a head and lose one and a half for a tail, and only five distinct totals turn out to be possible at all.17 min · 17 parts22 minAdding two probabilities double-counts the overlap, so the overlap comes back offAdding two probabilities overshoots, and the amount it overshoots by IS the overlap - not approximately, but on every one of the 32,640 pairs of events a three-coin list carries. That identity is the whole rule, and the explanation proves it twice, tests where plain adding is accidentally right, and shows that the consistency test everyone is taught is only half the test.22 min · 18 parts17 minKnowing one probability hands you its opposite for freeThe rule is one line long, which is exactly why it is worth taking apart. It uses two of the three axioms and not the third, it does not need the outcomes to be equally likely, and the standard ten-card illustration is chosen so that naming the complement by a property gets it wrong. All of that is measured here rather than asserted.17 min · 16 parts

What you will be able to do

  • State the chapter's definition of an event, and say why the word "any" in it cannot be weakened
  • Produce a description, for a stated experiment, whose subset is empty, and one whose subset is the whole sample space
  • Write the complement of a stated event as an explicit list, and check that the two lists together exhaust the sample space
  • Test a stated pair of events for a shared outcome, and conclude on that basis alone whether they can occur together
  • State the three conditions the chapter imposes, and say for each what it rules out
  • Compute an event's probability by adding the values on its outcomes, for an assignment that is not uniform
  • Show by a worked case that totalling two probabilities overshoots the union, and identify the overshoot as the overlap
  • Prove that an event and its complement have probabilities totalling 1, naming the two axioms used

What this chapter assumes you already know

  • Set membership, subsets, the empty set, and that a set is fixed by which elements belong to it
  • Union, intersection, difference and complement as operations on sets, from the chapter on sets earlier in this book
  • Listing every outcome of a small experiment systematically — two coin tosses, one die — without repeating or omitting one
  • Reading set-builder notation of the form "all elements with this property"
  • Assumed by the chapter but not supplied anywhere in this book: what a random experiment is, and how a sample space is built for one. See Notes.
  • 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

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "An event is one outcome."
  • "Different wording means different events."
  • "The empty set is not an event, it is the absence of one."
  • "An event occurred because it was the likely one."
  • "HT and TH are the same thing."
  • "An event has to be describable in a short sentence."
  • "The sample space is the biggest event, so it is not really one."
  • "Simple means easy to describe."