{"slug":"cbse-class-10-maths-probability","title":"Probability","description":"Theoretical probability, elementary, sure, impossible and complementary events, and problems with coins, dice and cards, weighted towards the complementary-event, two-coin and two-dice questions set across the 2026 papers.","board":"CBSE","grade":"Class 10","subject":"Maths","chapter":"Probability","url":"https://www.flipnlearn.app/deck/cbse-class-10-maths-probability","studyUrl":"https://www.flipnlearn.app/deck/cbse-class-10-maths-probability/study","cards":[{"id":"equally-likely-outcomes","kind":"term-definition","front":"Equally likely outcomes","back":"Outcomes that each have the same chance of happening, such as heads or tails on a fair coin, or 1 to 6 on a fair die. Drawing red or blue from a bag of 4 red and 1 blue ball is not equally likely."},{"id":"theoretical-probability","kind":"formula","front":"Theoretical probability","back":"P(E) = (outcomes favourable to E) ÷ (all possible outcomes), assuming all outcomes of the experiment are equally likely. It is also called classical probability."},{"id":"experimental-and-theoretical-probability","kind":"concept-question","front":"How does theoretical probability differ from experimental probability, and how are they related?","back":"Experimental probability is based on what actually happened over repeated trials; theoretical probability predicts from assumptions such as equally likely outcomes, without repeating anything. As the number of trials grows, the two are expected to come close."},{"id":"elementary-event","kind":"term-definition","front":"Elementary event","back":"An event consisting of just one outcome of an experiment, such as a head turning up in a single toss of a coin."},{"id":"sum-of-probabilities-of-elementary-events","kind":"formula","front":"Sum of the probabilities of all elementary events","back":"Adding the probabilities of every elementary event of an experiment always gives 1. For a red, a blue and a yellow ball, P(R) + P(B) + P(Y) = 1/3 + 1/3 + 1/3 = 1."},{"id":"complementary-event","kind":"term-definition","front":"Complementary event","back":"The event 'not E', written Ē, which happens exactly when E does not. E and Ē are complementary; for a die, 'a number greater than 4' and 'a number less than or equal to 4' are complements."},{"id":"sum-of-complementary-probabilities","kind":"formula","front":"Sum of complementary probabilities","back":"P(E) + P(not E) = 1, so P(not E) = 1 − P(E). It is often quicker to find the probability of the complement and subtract from 1."},{"id":"impossible-event","kind":"term-definition","front":"Impossible event","back":"An event that cannot happen, so no outcome is favourable to it and its probability is 0. Getting an 8 with a single throw of a die is an example."},{"id":"sure-event","kind":"term-definition","front":"Sure event","back":"An event certain to happen, so every outcome is favourable and its probability is 1. It is also called a certain event; getting a number less than 7 on one throw of a die is an example."},{"id":"range-of-probability","kind":"formula","front":"Range of probability","back":"0 ≤ P(E) ≤ 1 for any event E. The number of favourable outcomes can never be more than the total number of outcomes, so the ratio cannot exceed 1 or be negative."},{"id":"deck-of-playing-cards","kind":"term-definition","front":"A deck of playing cards","back":"52 cards in 4 suits of 13: spades and clubs are black, hearts and diamonds are red. Each suit runs ace, king, queen, jack and 10 down to 2, and kings, queens and jacks are called face cards."},{"id":"how-to-find-the-probability-of-drawing-an-ace","kind":"worked-step","front":"One card is drawn from a well-shuffled deck of 52. How do you find the probability that it is an ace, and that it is not?","back":"There are 4 aces among 52 equally likely cards, so P(ace) = 4/52 = 1/13. Not an ace is the complement, so P(not ace) = 1 − 1/13 = 12/13, which matches counting the 48 other cards."},{"id":"how-to-use-complements-when-one-player-must-win","kind":"worked-step","front":"If the probability that Sangeeta wins a tennis match against Reshma is 0.62, how do you find the probability that Reshma wins?","back":"Exactly one of them wins, so 'Reshma wins' is the complementary event of 'Sangeeta wins'. Therefore P(Reshma wins) = 1 − 0.62 = 0.38."},{"id":"how-to-find-the-probability-of-a-shared-birthday","kind":"worked-step","front":"Ignoring leap years, how do you find the probability that two friends have different birthdays, and the same birthday?","back":"Fix one friend's birthday; the other's can be any of 365 equally likely days, and 364 of them differ. So P(different) = 364/365, and by complements P(same) = 1 − 364/365 = 1/365."},{"id":"how-to-find-at-least-one-head-with-two-coins","kind":"worked-step","front":"Two different coins are tossed together. How do you find the probability of at least one head?","back":"List the equally likely outcomes: HH, HT, TH and TT. Three of the four contain a head, so the probability is 3/4. Or use the complement: P(no head) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4."},{"id":"how-to-find-probabilities-with-two-dice","kind":"worked-step","front":"Two dice, one blue and one grey, are thrown. How do you find the probability that the numbers add up to 8?","back":"List all ordered pairs in a 6 × 6 grid, giving 36 equally likely outcomes. The pairs summing to 8 are (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2), so the probability is 5/36. A sum of 13 is impossible, with probability 0."},{"id":"why-ordered-pairs-count-separately","kind":"concept-question","front":"When two dice are thrown, why are (1, 4) and (4, 1) counted as different outcomes?","back":"The dice are distinguishable, blue and grey, so '1 on blue, 4 on grey' and '4 on blue, 1 on grey' are separate results. Counting ordered pairs gives 36 equally likely outcomes; the 11 possible sums are not equally likely."},{"id":"why-grouped-outcomes-are-not-equally-likely","kind":"concept-question","front":"Why is it wrong to say two heads, two tails and one of each each have probability 1/3 when two coins are tossed?","back":"Those three results are not equally likely. 'One of each' can happen in two ways, HT and TH, while two heads and two tails can each happen in only one way. Out of four equally likely outcomes, one of each has probability 1/2."}]}