Probability is a measure of how likely an event is to occur. It is a number between 0 and 1 inclusive, where 0 means the event is impossible and 1 means the event is certain. In Class 10, we study theoretical (classical) probability.
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Key Definitions
Experiment: An action with an observable outcome (e.g., rolling a die).
Sample Space (S): The set of all possible outcomes. Its size is written as n(S).
Event (E): A subset of the sample space — one or more outcomes.
Favourable Outcomes: The outcomes that satisfy the event condition.
Theoretical Probability:
P(E) = (Number of favourable outcomes) / (Total number of outcomes) = n(E) / n(S)
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Key Properties
Key formulas
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Common Sample Spaces
Coin (fair): H, T. n(S) = 2.
Two coins: HH, HT, TH, TT. n(S) = 4.
Three coins: n(S) = 8.
Die (fair): 1, 2, 3, 4, 5, 6. n(S) = 6.
Two dice: n(S) = 36.
Pack of 52 cards: 4 suits (Spades, Hearts, Diamonds, Clubs), each with 13 cards (A, 2-10, J, Q, K). Face cards = J, Q, K (12 total). Aces = 4.
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Worked Examples
A fair coin is tossed. Find P(Head).
- n(S) = 2, favourable = {H}, n(E) = 1
- P(Head) = 1/2
A die is thrown once. Find P(getting a prime number).
- Primes on a die: 2, 3, 5. n(E) = 3.
- P = 3/6 = 1/2
Two coins are tossed simultaneously. Find P(getting at least one head).
- S = {HH, HT, TH, TT}. Favourable = {HH, HT, TH}. n(E) = 3.
- P = 3/4
A card is drawn from a pack of 52. Find P(getting a red king).
- Red kings = King of Hearts + King of Diamonds = 2.
- P = 2/52 = 1/26
A bag contains 3 red, 4 blue, and 5 green balls. A ball is drawn at random. Find P(not blue).
- Total = 12. Blue = 4. Not blue = 8.
- P(not blue) = 8/12 = 2/3
Two dice are rolled. Find P(sum = 7).
- Favourable pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). n(E) = 6.
- P = 6/36 = 1/6
From the integers 1 to 20, one number is picked at random. Find P(it is divisible by 3 or 5).
- Divisible by 3: 3,6,9,12,15,18 = 6 numbers.
- Divisible by 5: 5,10,15,20 = 4 numbers.
- Divisible by both (15): 1 number.
- Total favourable = 6 + 4 - 1 = 9.
- P = 9/20
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Common mistakes
> Common mistakes: Students sometimes list sample spaces incorrectly, especially for two dice or two coins. For two coins, HT and TH are different outcomes! Also, probability can never exceed 1 or be negative — if your answer is outside [0, 1], recheck. Do not add probabilities of overlapping events without subtracting the intersection (use inclusion-exclusion).
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Summary
Probability measures the likelihood of an event using the formula P(E) = n(E)/n(S). It always lies between 0 and 1. The complement rule P(E') = 1 - P(E) is very useful. For cards and dice, carefully list all favourable outcomes. Remember that for two dice, n(S) = 36, and for two coins, n(S) = 4.