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CBSE Class 9 · Mathematics

The Mathematics of Maybe: Introduction to Probability

Official NCERT chapter from Ganita Manjari Part I (book code iemh1). ExamMaster notes are original teaching at CBSE Class 9 depth.

This lesson follows the official chapter “The Mathematics of Maybe: Introduction to Probability” in Ganita Manjari Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 9
  • Medium level
  • 8 concepts

1What is Probability?

Probability here is a fair share of equally likely outcomes: how many named results fit the event, over how many named results the trial can have. A coin’s head is 1 of 2; a “maybe” feeling is not a number until the trial and the event are named.

Saying “probably” without a trial is a mood, not this measure.

Figure. Fourteen equally likely counters. Eight are red, so P(red) = 8/14. The widths 8:6 follow the counts.

How it works

  1. Name the trialA fair coin, a fair die — as given.
  2. Name the event as a subset of those outcomesHeads; or even.
  3. Write the share n(event)/n(all) when equally likelyThe maybe as a number.

2Measuring Probability Objectively

Objective measure means the share comes from the listed outcomes, not from how badly you want the event. A sure event uses every outcome (share 1); an impossible event uses none (share 0). Every event’s share sits between 0 and 1 inclusive.

Raising a share because you “need” the mark is not objective.

Figure. An equally-likely probability is a point on 0 to 1. 6 red out of 10 is 0.6, between impossible (0) and sure (1).

How it works

  1. List the equally likely outcomesThe sample.
  2. Count how many sit in the eventThe favourable.
  3. Write the fraction; 0 if none, 1 if allThe measure.

A fair die, even

A fair six-face die is rolled. What is the probability of an even face?

  • Outcomes1,2,3,4,5,6 — six equally likely
  • Even2,4,6 — three
  • Share3/6 = 1/2

Pro tip. Count first; cancel only after the counts are honest.

3Elements of Probability: Sample Spaces and Events

A sample space is the named list of what the trial can give. An event is a named sub-list. “Even” is not a sample space; {1,2,3,4,5,6} is. Two events can share outcomes; write the overlap if the question asks a union or an “and”.

A feeling-list (“lucky numbers”) is not a sample space.

Figure. S is the six faces. E is the even faces 2, 4, 6. P(E) = n(E)/n(S) = 3/6. Do not count a face twice.

How it works

  1. Write every outcome the trial allowsS.
  2. Circle the event’s membersE.
  3. Keep S as the whole list, E as a partElements of the maybe.

4Tree diagrams

A tree diagram splits a two-step trial: first toss, then second. Each path is one outcome of the pair. Two fair coins have four paths HH, HT, TH, TT. The probability of a path is the product of the step-shares when the steps are independent as taught.

A tree with missing branches under-counts S.

Figure. Two fair coins: four equally likely leaves HH, HT, TH, TT. Each path is 1/2 then 1/2. P(one head) = 2/4.

How it works

  1. Draw the first-step branchesH or T.
  2. From each, draw the second-step branchesThe four paths.
  3. Count paths in the event over all pathsThe tree did the listing.

5A definition is a test you can run

Probability is a test: a named trial, a listed S, an event as a subset, a share in [0, 1]. If you cannot name S, you have a mood-word.

A pretty “50-50” badge without a list is not the test.

Figure. The definition-test: P(E) + P(not E) must be 1. For 5 red and 3 blue, 5/8 + 3/8 = 1. If the two parts miss 1, a count was double-counted or dropped.

How it works

  1. Name the trial and list SThe object.
  2. Name E and write n(E)/n(S) when equally likelyThe test.
  3. Then the word has a numberThe definition ran.

6Name the given before the unknown

The given is the trial and what “success” means. The unknown is the share. Copy the faces or the paths before you cancel.

Calling a die “even” and then counting {2,4} only is a silent given (you dropped 6).

Figure. Name the given counts before the unknown: 8 red, 6 blue, total 14. Only then write P(red) = 8/14.

How it works

  1. Copy the trial and the event-wordsThe given.
  2. Name the shareThe unknown.
  3. Then list and countGiven first.

7One worked case is enough at this class

One trial-and-share is enough. A page of the same 3/6 clone does not add a new sample space.

Ten twins of a fair coin are still one 1/2.

Figure. One worked bag: 6 red and 4 blue, total 10. P(red) = 6/10. Bar length is the count, not the probability word.

How it works

  1. Run one listed S and one eventThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A cricket-match photo is a setting. The maths is a listed S and a share. Cheering is not n(E).

A stadium still is not a sample space.

Figure. The miss is swapping the count for the probability. 8 is how many red counters; 8/14 is P(red). A name is not the ratio.

How it works

  1. Keep the match as a storyA setting.
  2. Write the trial you were actually given (coin, die, two coins)The maths.
  3. Then countDo not swap.
A fair coin’s P(heads) is
  1. 1/2 — one of two equally likely faces
  2. 2
  3. 0 because it might be tails

Sure is 1; a single face of two is 1/2.

Notes

  • The official chapter title is “The Mathematics of Maybe: Introduction to Probability”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • P(E) = n(E)/n(S) when outcomes are equally likely
  • 0 ≤ P(E) ≤ 1

Recap

Hold these pegs from the official chapter “The Mathematics of Maybe: Introduction to Probability”. The wording is ExamMaster’s teaching, not a textbook recap.

What is Probability?
Probability here is a fair share of equally likely outcomes: how many named results fit the event, over how many named results the trial can have.
Measuring Probability Objectively
Objective measure means the share comes from the listed outcomes, not from how badly you want the event.
Elements of Probability: Sample Spaces and Events
A sample space is the named list of what the trial can give.
Tree diagrams
A tree diagram splits a two-step trial: first toss, then second.
A definition is a test you can run
Probability is a test: a named trial, a listed S, an event as a subset, a share in [0, 1].
Name the given before the unknown
The given is the trial and what “success” means.

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