13. Probability

7 topics2 h

Chapter 13 of NCERT Mathematics for Class 12: Probability. Three sections — Probability after you learn something, Chaining and independence and Reversing the conditioning. Seven videos, 2 h in all.

Worked answers to the 62 exercise questions in this chapter: Exercise 13.1 · Exercise 13.2 · Exercise 13.3 · Miscellaneous Exercise

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Part 1

Probability after you learn something

14 minBeing told one event happened shrinks the sample space you count against14 min · 13 parts15 minThe rules conditional probability inherits, including addition on a restricted space15 min · 13 parts

Part 2

Chaining and independence

15 minMultiplying along a chain of dependent draws, and extending it past two events15 min · 12 parts19 minIndependence, and why it is a different idea from having no outcome in common19 min · 14 parts

Part 3

Reversing the conditioning

16 minPartitioning the sample space, and totalling a probability across the parts16 min · 12 parts17 minWorking back from an observed effect to the likelihood of each cause17 min · 13 parts13 minAttaching a number to every outcome: a random variable as a function on the sample space13 min · 11 parts

What you will be able to do

  • Describe in one sentence what being told that an event has happened does to the list of outcomes still in play
  • State the claim that a conditional probability is itself a probability, and say what would have to fail for that to be false
  • Derive the product form from the definition of a conditional probability, in one line, in both orders
  • Compute both conditionals for a suit-and-rank pair on a deck and observe that each equals the corresponding unconditional value
  • State the question §13.5 opens with, and say which direction of conditioning is already available and which is not
  • State the reversal formula with all of its hypotheses
  • State what kind of object a random variable is, in terms of domain and outputs

What this chapter assumes you already know

  • Sample space, outcome and event from Class XI, and an event written as a set
  • Equally likely outcomes, and probability as a count of favourable cases over a count of all cases
  • The axiomatic treatment of probability, which the chapter's opening page names as prior learning
  • Union, intersection and complement of sets, and the empty set
  • Reading a set of ordered pairs or ordered triples as a list of outcomes
  • Adding fractions with unlike denominators, and dividing one fraction by another
  • The definition of a conditional probability from the previous topic, including its side condition
  • The addition rule for two events on an unrestricted space, from Class XI

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.

  • "Conditioning changes the coins."
  • "The probability of the first given the second is the same as the second given the first."
  • "Divide by the probability of the event you want."
  • "The condition on the denominator is a technicality."
  • "Conditional probability is a smaller probability."
  • "Both children being boys, given one is a boy, is one half."
  • "Once outcomes stop being equally likely, the whole method fails."
  • "Every branch label on a tree is a branch chance."