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CBSE Class 10 · Mathematics

Probability

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Probability” in Mathematics. 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 10
  • Medium level
  • 8 concepts

1Probability — A Theoretical Approach

Theoretical probability (equally likely outcomes) is n(E)/n(S). A fair die’s even face is 3/6. Theoretical means the share from the listed equally likely set, not from ten rushed trials that happened to miss 6.

A “feeling 80%” is not this approach.

Figure. Theoretical probability lives on the unit interval. A fair coin lands at 1/2 because one of two equally likely faces is favourable.

How it works

  1. List S as equally likelyThe theory’s given.
  2. Count E inside SThe favourable.
  3. Write n(E)/n(S)The theoretical share.

2An outcome is one possible result

An outcome is one possible result: 4 on the die, HT on two coins. It is an atom of S, not an event-word like “even” (that event holds three outcomes). Outcome is one ticket; event is a bundle.

Calling “even” an outcome hides 2, 4, and 6.

Figure. An outcome is one finished face of the experiment. The 4 is one of six equally likely results, not a group of results.

How it works

  1. Name one result the trial can giveThe outcome.
  2. Keep events as sets of thoseThe bundle.
  3. Refuse a mood-word as a single outcomeOne ticket.

3Probability is favourable over possible, when equally likely

Probability is favourable over possible when equally likely — the same share as Class 9, now named theoretical. Cancel only after the counts are honest. 3/6 = 1/2, not “3 out of 5 because 6 felt unlucky”.

Dropping an outcome you dislike is a silent S.

Figure. When every counter is equally likely, probability is the favourable count over the whole list: 5 red among 8 counters.

How it works

  1. Count favourable, count possibleThe two numbers.
  2. Write the fractionThe probability.
  3. Cancel after the countsHonest share.

Even on a fair die

Theoretical P(even) on a fair six-face die.

  • Ssix faces
  • E2,4,6 — three
  • P3/6 = 1/2

Pro tip. Favourable over possible; cancel after the counts.

4A certain event is 1; an impossible event is 0

A certain event uses every outcome (P = 1); an impossible event uses none (P = 0). Every event sits in [0, 1]. Certain is “the whole list”, not “I am sure”.

Raising a share to 1 because you need the mark is not certain.

Figure. A certain event fills the sample space and scores 1. An impossible event is empty and scores 0. Every other event sits strictly between.

How it works

  1. See whether E is all of S, none, or someThe size.
  2. Write 1, 0, or the fractionThe measure.
  3. Keep 0 ≤ P ≤ 1The bounds.

5Complementary events add to 1

Complementary events split S: E and “not E”. Their shares add to 1. P(not even) = 1 − 1/2 = 1/2 on the die. Complement is the rest of the list, not a rival trial.

Adding P(even) to P(odd) and also to P(prime) without care double-uses 2,3,5.

Figure. E and not-E partition the sample space. Their probabilities add to 1 because every outcome sits in exactly one of the two blocks.

How it works

  1. Name EThe event.
  2. Write P(not E) = 1 − P(E)The complement.
  3. Check they partition SAdd to 1.

6Independent events multiply

Independent events (as taught) multiply: two fair coins, P(both heads) = (1/2)×(1/2) = 1/4. Independent means one trial does not change the other’s share. A without-replacement draw is usually not this multiply unless the lesson framed it that way.

Multiplying because it “looks advanced” on a single die is a miss.

Figure. Two independent tosses multiply: each path is one half times one half. The four pairs HH, HT, TH, TT are the equally likely list.

How it works

  1. See two steps that do not change each otherIndependent as taught.
  2. Multiply the sharesThe and.
  3. A tree lists the same 1/4 as one path of fourThe check.

7A tree lists the path without double counting

A tree lists each path without double counting: first step, then second. Two coins: four paths. The tree is the list of S for a two-step trial. A missing branch under-counts possible.

A tree that joins HT and TH as one path under-counts.

Figure. A tree lists each path once. HH is the H-then-H branch, not a second count of H. Multiply along a path; do not add the same toss twice.

How it works

  1. Draw first-step branchesThe split.
  2. From each, draw the nextThe paths.
  3. Count paths in E over all pathsNo double count.

8Mutually exclusive events add when they cannot both happen

Mutually exclusive events cannot both happen: even and odd on one roll. Then P(E or F) = P(E)+P(F). Exclusive is “no shared outcome”. Even and “prime” share 2 — not exclusive, so a raw add double-counts 2.

Adding any two events you like is not this rule.

Figure. Odd and even on a die cannot both happen on one toss, so P(odd or even) is the sum 1/2+1/2. Add only when the blocks share no outcome.

How it works

  1. See whether E and F share an outcomeThe overlap-test.
  2. If none, add the sharesExclusive or.
  3. If they share, subtract the overlap if the lesson taught that, or list SHonest.
P(even) + P(not even) on a fair die is
  1. 1 — complements
  2. 1/2
  3. 2

The rest of the list; they add to 1.

Notes

  • The official chapter title is “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) equally likely
  • P(not E)=1−P(E)
  • exclusive: P(E or F)=P(E)+P(F)

Recap

Hold these pegs from the official chapter “Probability”. The wording is ExamMaster’s teaching, not a textbook recap.

Probability — A Theoretical Approach
Theoretical probability (equally likely outcomes) is n(E)/n(S).
An outcome is one possible result
An outcome is one possible result: 4 on the die, HT on two coins.
Probability is favourable over possible, when equally likely
Probability is favourable over possible when equally likely — the same share as Class 9, now named theoretical.
A certain event is 1; an impossible event is 0
A certain event uses every outcome (P = 1); an impossible event uses none (P = 0).
Complementary events add to 1
Complementary events split S: E and “not E”.
Independent events multiply
Independent events (as taught) multiply: two fair coins, P(both heads) = (1/2)×(1/2) = 1/4.

Practise Probability

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