7. The Mathematics of Maybe: Introduction to Probability

10 topics2 h

Chapter 7 of NCERT Mathematics for Class 9: The Mathematics of Maybe: Introduction to Probability. Four sections — What probability measures, Two honest ways to get a number, Sample spaces and events and Multi-step experiments. Ten videos, 2 h in all.

Worked answers to the 28 exercise questions in this chapter →

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Part 1

What probability measures

9 minProbability as a measurement, not a guessTwo friends look at the same sky and predict opposite weather. Neither has made a mistake — and that is the problem this chapter exists to solve.9 min · 12 partsहिंदी9 minWhat we mean by randomRandom is a much narrower word than it sounds. You can write the whole list down; what you cannot do is say which entry turns up.9 min · 12 partsहिंदी9 minThe 0-to-1 scale, and what the endpoints meanSomebody says your chance of winning is 0.75. Nobody chose 0 and 1 as the ends of that scale — counting a part of a whole forces them.9 min · 12 partsहिंदी

Part 2

Two honest ways to get a number

10 minExperimental probability: relative frequency over many trialsFifty rolls of a fair die will not give exactly one sixth. They cannot: a sixth of fifty is not a whole number.10 min · 12 partsहिंदी10 minTheoretical probability: counting favourable outcomes when all are equally likelyFavourable over possible is not what probability means. It is a result that holds under one condition, and the condition carries all the content.10 min · 12 partsहिंदी11 minEstimating from statistical data, and scaling the estimate upTurning a survey into a claim about a whole school is one multiplication. Everything difficult about it is in the assumption underneath.11 min · 12 partsहिंदी10 minFair, unbiased, and memoryless: the gambler's fallacyAfter six heads, tails is not due. The proportion settles by dilution, not by correction — a surplus is divided, never repaid.10 min · 12 partsहिंदी

Part 3

Sample spaces and events

10 minListing every outcome: the sample spaceA sample space is chosen, not discovered. The same tomorrow can be handled with two outcomes or with four, and the question decides which.10 min · 12 partsहिंदी10 minAn event is a selection from the sample spaceAn event is a subset, and that one sentence turns an English question into arithmetic. “At least one head” is where it goes wrong.10 min · 12 partsहिंदी

Part 4

Multi-step experiments

10 minTree diagrams make the sample space of a two-step experiment visibleA tree is not a tidier way of writing the list. It is two multiplications drawn — and branch numbers add at a split but multiply along a path.10 min · 12 partsहिंदी

What you will be able to do

  • State what probability measures, and name the three physical quantities the chapter uses as its analogy
  • State the chapter's definition of randomness in terms of a known outcome list and an unpredictable outcome
  • State the range probability is measured on, and what a value of 0 and a value of 1 each mean
  • State the two objective routes §7.2 offers, and say what evidence each one uses
  • State the theoretical-probability formula and the assumption it requires
  • Compute a probability estimate from survey counts as a relative frequency
  • State the gambler's fallacy and identify it in a described piece of reasoning
  • Write the sample space of a described experiment inside braces, and state its size n(S)
  • State what an event is, and write a described event as a selection from a named sample space
  • State what a multi-step experiment is, using the chapter's own definition

What this chapter assumes you already know

  • Reading a decimal between 0 and 1, and converting between a decimal, a fraction and a percentage
  • The idea of a quantity that gets a number attached to it by a rule — length, area, volume from earlier classes
  • Listing all the ways a simple situation can turn out (a match won, drawn or lost)
  • That "40% of 1500" is a multiplication, needed only for the closing pointer to §7.2.3
  • Writing out every way a simple situation can turn out
  • Reading 1/2 as a half and as 50%
  • That a fraction of the form 1/n gets smaller as n grows — needed only to see why a school-wide lucky draw gives each student a small chance
  • Converting between a fraction, a decimal and a percentage: 3/4, 0.75, 75%

It builds directly on Placing a rational number on the line, and distance as |a − b|, Why long division must either stop or loop and What "rational" means, and why the denominator cannot be zero.

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.

  • "Probability tells you what will happen."
  • "If two people give different probabilities, one of them has made a mistake."
  • "Subjective probability is worthless."
  • "50% means I have no idea."
  • "A hockey match has two outcomes."
  • "Probability is about gambling."
  • "Because probability is a measurement, it must be exact."
  • "Random means anything at all could happen."