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CBSE Class 12 · Mathematics

Probability

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Probability” in Mathematics Part I–II. 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 12
  • Medium level
  • 8 concepts

1Conditional Probability

Conditional probability: P(E|F)=P(E∩F)/P(F) when P(F)>0 as taught. “Given F” shrinks the sample. If P(E∩F)=1/6 and P(F)=1/2, then P(E|F)=1/3. Conditional is a given-F write, not a second independent-guess.

Using P(E)/P(F) without the intersection is a miss.

Figure. Conditional probability keeps only B as the new whole. The favourable piece is the overlap A and B, so P(A given B) is that overlap over B, not over S.

How it works

  1. Name E and the given FThe pair.
  2. Form P(E∩F)/P(F)Conditional.
  3. Keep P(F)>0The gate.

Given F

P(E∩F)=1/6, P(F)=1/2. Find P(E|F).

  • WriteP(E∩F)/P(F)
  • Compute(1/6)/(1/2)=1/3
  • Readgiven F, chance of E is 1/3

Pro tip. Intersection over the given.

2Multiplication Theorem on Probability

Multiplication theorem: P(E∩F)=P(F)P(E|F) as taught (or P(E)P(F|E)). It is the rearrange of conditional. Use it when a “and then” chain is given.

Multiplying P(E)P(F) always is the independent-steal.

Figure. The multiplication rule follows a path: P(A and B) = P(A) times P(B given A). Here that is (1/2) times (2/5) = 1/5. The second factor is not P(B) unless the events are independent.

How it works

  1. Write the chain as an intersectionThe and.
  2. Factor as P(first)×P(second|first)The theorem.
  3. Keep independence for the next headingA special case.

3Independent Events

Independent events: P(E∩F)=P(E)P(F) as taught. Independence is a multiply-test, not “they look unrelated”. If P(E)=1/3, P(F)=1/2 and they are independent, P(E∩F)=1/6. Independent is not disjoint (those have empty intersection).

Calling disjoint events independent is a common miss — disjoint with positive P cannot be independent.

Figure. Two fair coins are independent: P(two heads) is (1/2) times (1/2), which is also the one cell in four. If a first head changed the second coin, the four cells would not stay equal.

How it works

  1. Test P(E∩F)=P(E)P(F)Independent.
  2. Keep disjoint as a different wordEmpty and.
  3. Refuse “unrelated looking” as the testA number-test.

4Bayes' Theorem

Bayes: reverse a conditional as taught — P(F|E) from P(E|F) and the prior P(F). A partition into F and F' (or more) if named. Bayes is a reverse-given, not a new definition of P. One two-box case is enough.

Swapping P(E|F) for P(F|E) without the rewrite is the leftover Bayes names.

Figure. Bag I has 3 red in 5; Bag II has 1 red in 5; bags equally likely. P(red) = 3/10 + 1/10 = 2/5. Bayes reverses the tree: P(Bag I given red) = (3/10) / (2/5) = 3/4.

How it works

  1. Write the partition and the forward P(E|F)The given.
  2. Form the taught Bayes writeThe reverse.
  3. Keep prior×likelihood in the numeratorHonest.

5A definition is a test you can run

Probability-word is a test: a given-F, a multiply-chain, independence, or Bayes. If you only say “chance”, you have a heading.

A dice-sticker is not the test.

Figure. A probability is a place on the unit interval. Zero is impossible, one is certain, and every school event sits between, including this P(E).

How it works

  1. Name conditional, product, independent, or BayesThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

6Name the given before the unknown

The given is two probabilities and a “given”. The unknown is the conditional or the reverse. Copy P(F)=1/2 before you divide by 1/3.

Using P(E) as the denominator of P(E|F) is a silent swap.

Figure. Name the given before the unknown. Once B is given, B becomes the whole sample space and A and B is the only favourable piece left.

How it works

  1. Copy P(E∩F) and P(F)The given.
  2. Name P(E|F) or the Bayes reverseThe unknown.
  3. Then divideGiven first.

7One worked case is enough at this class

One worked case is enough: P(E|F)=1/3. Do not stack four Bayes boxes. The case teaches intersection-over-given.

A poster of ten trees is not more science.

Figure. One bag, 8 red and 6 blue, equally likely draws. Favourable over possible is 8/14. Do not invent a second draw or a condition this case did not give.

How it works

  1. Take one conditionalThe case.
  2. Divide intersection by P(F)The teach.
  3. Stop — one caseThis class.

8A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Independent” is P-and equals product — a two-dice glow is a setting.

A pretty tree without a product-test is a mute picture.

Figure. The miss is swapping the given. Both fractions share the same numerator A and B; the denominators are different events, so the two conditionals are not interchangeable.

How it works

  1. Keep conditional/independent/Bayes as the namesThe science.
  2. Use a tree only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.
If P(E∩F)=1/6 and P(F)=1/2, then P(E|F) is
  1. 1/3
  2. 1/6
  3. 1/12

Intersection over given.

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|F)=P(E∩F)/P(F)
  • P(E∩F)=P(E)P(F) if independent
  • Bayes: reverse the given as taught

Recap

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

Conditional Probability
Conditional probability: P(E|F)=P(E∩F)/P(F) when P(F)>0 as taught.
Multiplication Theorem on Probability
Multiplication theorem: P(E∩F)=P(F)P(E|F) as taught (or P(E)P(F|E)).
Independent Events
Independent events: P(E∩F)=P(E)P(F) as taught.
Bayes' Theorem
Bayes: reverse a conditional as taught — P(F|E) from P(E|F) and the prior P(F).
A definition is a test you can run
Probability-word is a test: a given-F, a multiply-chain, independence, or Bayes.
Name the given before the unknown
The given is two probabilities and a “given”.

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