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Bihar Board Class 10 · Maths · Matric exam 2027

Chapter 15: Probability — notes

The chapter’s 6 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.

This chapter’s important questions →

  1. 1. Experimental vs Theoretical Probability

    Okay, so you already know this from Class IX — experimental probability is based on what actually happens when you do something many times, again and again.

    • • Empirical P(E) = (trials where E happened)/(total trials)
    • • Coin tossed 1000 times: Head 455, Tail 545
    • • P(head) = 455/1000 = 0.455, P(tail) = 0.545
    • • Buffon: 4040 tosses, 2048 heads, P = 0.507
    • • Kerrich: 10000 tosses, 5067 heads, P = 0.5067
    • • Pearson: 24000 tosses, 12012 heads, P = 0.5005

  2. 2. Equally Likely Outcomes

    Okay, so let's talk about equally likely outcomes.

    • • Coin toss: outcomes Head, Tail — equally likely
    • • Die throw: outcomes 1,2,3,4,5,6 — equally likely
    • • Bag: 4 red, 1 blue ball — red and blue NOT equally likely
    • • We assume equally likely outcomes from now on

  3. 3. Theoretical Probability and the Formula

    Okay, so now we come to the main definition of this chapter.

    • • P(E) = (outcomes favourable to E)/(all possible outcomes)
    • • Given by Pierre Simon Laplace, 1795
    • • Coin: P(head) = 1/2, P(tail) = 1/2
    • • Bag with 1 red,1 blue,1 yellow: P(Y) = 1/3, P(R) = 1/3, P(B) = 1/3
    • • Elementary event: only one outcome favourable
    • • Sum of all elementary event probabilities = 1

  4. 4. Complementary Events: P(E) + P(not E) = 1

    Okay, so let's go back to the die.

    • • E = getting number >4 on die: P(E) = 2/6 = 1/3
    • • F = getting number ≤4: P(F) = 4/6 = 2/3
    • • P(E)+P(F) = 1, so F is 'not E'
    • • P(E) + P(not E) = 1
    • • P(not E) = 1 − P(E)
    • • Sangeeta P(S) = 0.62, Reshma P(R) = 1-0.62 = 0.38

  5. 5. Impossible and Sure Events; the Range of Probability

    Okay, so think about throwing a die once.

    • • P(getting 8 on a die) = 0/6 = 0 → impossible event
    • • P(getting number <7 on a die) = 6/6 = 1 → sure event
    • • 0 ≤ P(E) ≤ 1 always

  6. 6. Working with Cards, Boxes and Two Dice

    Okay, so let's apply the same formula to a few more situations.

    • • Deck of 52 cards: 4 suits × 13 cards
    • • P(ace) = 4/52 = 1/13; P(not ace) = 48/52 = 12/13
    • • Box: 3 blue+2 white+4 red = 9 marbles
    • • P(white) = 2/9, P(blue) = 1/3, P(red) = 4/9
    • • Two coins: outcomes (H,H)(H,T)(T,H)(T,T), P(at least one head) = 3/4
    • • Two dice: 36 outcomes; P(sum = 8) = 5/36; P(sum = 13) = 0; P(sum≤12) = 1

Summary

Okay, so let's go back over this whole chapter. We started with two kinds of probability. One is experimental — you actually toss the coin, or roll the die, and count what happens. The other is theoretical — you reason it out, using outcomes that are equally likely. We looked at what "equally likely" really means, with a coin, a die, and a bag of balls. That gave us our main formula, from Laplace — P(E) = number of outcomes favourable to E / number of all possible outcomes. We used this with coins, balls, cards, marbles, students, dice — lots of examples. From there, you saw that an event with just one outcome is called an elementary event, and all the elementary events together add up to a probability of 1. You also saw P(E) + P(not E) = 1, for complementary events. An impossible event has probability 0. A sure event has probability 1. And every probability you ever find lies between 0 and 1 — never outside it. So, tell me — can you think of one event in daily life that has probability 1?

AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 15. Written from the textbook, checked twice.

Questions parents and students ask

How do I get the "Probability" notes as a PDF?
Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
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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