CBSE Class 10 · Maths · board exam 2027
Chapter 14: Probability — notes
The chapter’s 10 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. From experimental to theoretical probability
In Class 9 you studied experimental probability.
- • Experimental probability = (number of trials in which the event happened)/(total number of trials)
- • Coin tossed 1000 times: head 455 times, tail 545 times
- • 455 + 545 = 1000
- • P(head) = 455/1000 = 0.455, P(tail) = 545/1000 = 0.545
- • Experimental probability = an estimate (it can change when the experiment is repeated)
- • | Tossed by | Tosses | Heads | P(head) |
- • | Buffon | 4040 | 2048 | 0.507 (about) |
- • | Kerrich | 10000 | 5067 | 0.5067 |
- • | Pearson | 24000 | 12012 | 0.5005 |
- • 0.5 = 1/2
- • Theoretical (classical) probability: no experiment, only reasoning
2. Outcomes and equally likely outcomes
First, two words.
- • Coin: outcomes head (H), tail (T)
- • Same chance ⇒ equally likely outcomes
- • Die: outcomes 1, 2, 3, 4, 5, 6 ⇒ 6 in all (equally likely)
- • Bag: 4 red balls, 1 blue ball
- • 'red' and 'blue' are NOT equally likely (4 red, only 1 blue)
- • But each ball has the same chance ⇒ 5 equally likely outcomes
- • In this chapter: outcomes are equally likely
3. Theoretical probability: the formula
The main formula of this chapter.
- • P(E) = (number of outcomes favourable to E)/(number of all possible outcomes)
- • (all outcomes equally likely)
- • ① Write all possible outcomes, count them
- • ② Pick out the favourable outcomes one by one, count them
- • ③ Make the fraction, write it in simplest form
- • Example 1: a coin tossed once
- • All outcomes: H, T ⇒ 2
- • P(head) = 1/2, P(tail) = 1/2
- • Example 2: bag with 1 red, 1 blue, 1 yellow ball
- • All outcomes: red, blue, yellow ⇒ 3
- • P(yellow) = 1/3, P(red) = 1/3, P(blue) = 1/3
- • Example 3: die, E = a number greater than 4
- • All outcomes: 1, 2, 3, 4, 5, 6 ⇒ 6
- • Favourable: 5, 6 ⇒ 2
- • P(E) = 2/6 = (2 ÷ 2)/(6 ÷ 2) = 1/3
4. Elementary events and complementary events
An event that has only one outcome is called an elementary event.
- • Elementary event = an event with only one outcome
- • Coin: 'getting a head', 'getting a tail'
- • P(head) + P(tail) = 1/2 + 1/2 = 1
- • 1/3 + 1/3 + 1/3 = 3/3 = 1
- • Sum of probabilities of all elementary events = 1
- • E = a number greater than 4 ⇒ P(E) = 1/3
- • F = a number less than or equal to 4: 1, 2, 3, 4 ⇒ 4
- • P(F) = 4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3
- • P(E) + P(F) = 1/3 + 2/3 = 3/3 = 1
- • F = 'not E' = Ē (E bar)
- • P(E) + P(Ē) = 1
- • P(Ē) = 1 − P(E)
- • E and Ē: complementary events
- • Example: P(Sangeeta wins) = 0.62
- • Reshma wins = 'Sangeeta does not win'
- • P(Reshma wins) = 1 − 0.62 = 0.38
- • P(different birthdays) = 364/365
- • P(same birthday) = 1 − 364/365 = 365/365 − 364/365 = 1/365
5. Impossible event, sure event and the range of probability
A die is thrown once.
- • Die: P(getting 8) = 0/6 = 0
- • Impossible event: P(E) = 0
- • Die: P(a number less than 7) = 6/6 = 1
- • Sure (certain) event: P(E) = 1
- • 0 ≤ favourable outcomes ≤ all possible outcomes
- • 0 ≤ P(E) ≤ 1
- • 1.2 > 1 ✗, −1.5 < 0 ✗
- • 15% = 15/100 = 0.15 ✓
6. Balls and marbles: drawing at random
One phrase comes again and again in board questions, at random.
- • At random = without looking, without choosing ⇒ every marble equally likely
- • Box: 3 blue, 2 white, 4 red marbles
- • Total marbles = 3 + 2 + 4 = 9
- • P(white) = 2/9
- • P(blue) = 3/9 = (3 ÷ 3)/(9 ÷ 3) = 1/3
- • P(red) = 4/9
- • 2/9 + 3/9 + 4/9 = 9/9 = 1
- • Not red = blue or white: 3 + 2 = 5 ⇒ P = 5/9
- • By complement: P = 1 − 4/9 = 9/9 − 4/9 = 5/9
7. A deck of 52 playing cards
Playing cards.
- • Deck of playing cards: 52 cards
- • 4 suits: spades ♠, hearts ♥, diamonds ♦, clubs ♣
- • 13 cards in each suit ⇒ 4 × 13 = 52
- • Black (spades, clubs) = 13 + 13 = 26
- • Red (hearts, diamonds) = 13 + 13 = 26
- • Each suit: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king
- • 1 + 9 + 3 = 13
- • Face cards = king, queen, jack
- • Face cards = 3 × 4 = 12
- • Kings 4, queens 4, jacks 4, aces 4
- • Example: one card from a well-shuffled deck
- • All outcomes = 52
- • P(ace) = 4/52 = (4 ÷ 4)/(52 ÷ 4) = 1/13
- • Not an ace: 52 − 4 = 48 ⇒ P = 48/52 = (48 ÷ 4)/(52 ÷ 4) = 12/13
- • By complement: 1 − 1/13 = 13/13 − 1/13 = 12/13
- • P(face card) = 12/52 = (12 ÷ 4)/(52 ÷ 4) = 3/13
8. Two coins and three coins
Toss two coins together.
- • Two coins: all outcomes = 2 × 2 = 4
- • HH, HT, TH, TT
- • HT ≠ TH (the first coin is different)
- • Both heads: HH ⇒ 1
- • P(two heads) = 1/4
- • At least one head: HH, HT, TH ⇒ 3
- • P(at least one head) = 3/4
- • Three coins: all outcomes = 2 × 2 × 2 = 8
- • HHH, HHT, HTH, HTT
- • THH, THT, TTH, TTT
- • Exactly two heads: HHT, HTH, THH ⇒ 3
- • P(exactly two heads) = 3/8
- • No head: TTT only ⇒ 1
- • P(at least one head) = (8 − 1)/8 = 7/8 = 1 − 1/8
9. Two dice: 36 outcomes
Two dice, one blue and one grey, thrown together.
- • Two dice: all outcomes = 6 × 6 = 36
- • (first number on the blue die, second number on the grey die)
- • | Blue ↓ Grey → | 1 | 2 | 3 | 4 | 5 | 6 |
- • | 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
- • | 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
- • | 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
- • | 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
- • | 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
- • | 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
- • (1,2) ≠ (2,1)
- • Probability of sum 8
- • Blue shows 1 ⇒ grey must show 7 ✗
- • 8 = 2 + 6 = 3 + 5 = 4 + 4 = 5 + 3 = 6 + 2
- • Favourable: (2,6), (3,5), (4,4), (5,3), (6,2) ⇒ 5
- • P(sum 8) = 5/36
- • Biggest sum = 6 + 6 = 12
- • P(sum 13) = 0/36 = 0
- • P(sum ≤ 12) = 36/36 = 1
- • Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) ⇒ 6
- • P(sum 7) = 6/36 = (6 ÷ 6)/(36 ÷ 6) = 1/6
10. Working backwards: from a probability to a number
Many board questions go the other way.
- • P(E) = (favourable outcomes)/(all outcomes)
- • ⇒ favourable outcomes = P(E) × all outcomes
- • Example: 20 balls, P(red) = 1/4
- • Red balls = 1/4 × 20 = 20/4 = 5
- • Green balls = 20 − 5 = 15
- • Check: P(green) = 1 − 1/4 = 3/4, 3/4 × 20 = 15 ✓
- • Unknown number ⇒ let it be x, make an equation, solve
- • Number of balls: a whole number, never negative
Summary
Very good! You have finished the whole chapter today. First we saw that as the trials grow, the experimental probability comes closer to the theoretical probability. Then we learnt the main formula, favourable outcomes upon all possible outcomes. Then complementary events, P(E bar) equals 1 minus P(E). The probability of an impossible event is 0, of a sure event is 1, and every probability lies from 0 to 1. And at the end we did marbles, the 52 playing cards, coins, the 36 outcomes of two dice, and finding a number from a probability. Write all the outcomes and count them, pick out the favourable outcomes one by one, then write the fraction in its simplest form. Today's homework is on your screen. Make the table of 36 outcomes for two dice in your notebook once, on your own. If anything was not clear, press the Ask button and ask me. Done, you worked very well today!
AskMedha · Medha Ma’am — CBSE Class 10 Maths, Chapter 14. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Probability" notes as a PDF?
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- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
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