Probability
CBSE Class 10 · Mathematics · Notes, formulas and practice questions
Theoretical probability for Class 10: measure the chance of an event by comparing favourable outcomes to equally likely total outcomes, with applications to coins, dice, playing cards and marbles.
Studying Probability?
Turn your own textbook pages, a lecture recording or a YouTube video into notes like these — plus flashcards, quizzes and practice questions — in your own language. The Standard plan is free, permanently, and needs no card.
Make notes from your own material — freeWhat this chapter covers
Probability measures the chance that a particular event will happen when an experiment is performed. In Class 10, we study theoretical probability, which is based on equally likely outcomes. For example, when we toss a fair coin, head and tail have equal chances. The value of a probability is always between 0 and 1. If an event never occurs, its probability is 0; if it always occurs, its probability is 1.
The basic formula compares favourable outcomes to total equally likely outcomes. For an experiment with many outcomes, we first list or count the total possible outcomes and then the outcomes that favour the event. This idea is applied to coins, dice, cards and marbles. For two coins or two dice, remember that each pair is an ordered outcome, so (head, tail) and (tail, head) are different. This careful counting is the most common place for errors.
The complement of an event E, written as not E, is the set of outcomes where E does not happen. Since every outcome either favours E or favours not E, the two probabilities add to 1. Thus P(E) + P(not E) = 1. This is useful when P(not E) is easier to find, for example in finding the probability of at least one head by subtracting the probability of no head.
Playing cards give many practice examples. A standard deck has 52 cards in four suits: hearts, diamonds, clubs and spades. Hearts and diamonds are red; clubs and spades are black. Each suit has 13 cards, including one ace and three face cards (king, queen, jack). Such structure allows us to count favourable outcomes without listing the whole deck. Marble problems are similar; we only need the number of marbles of each colour.
Key terms
- Random experiment
- An action or process whose result cannot be predicted before it is performed. Tossing a coin, rolling a die or drawing a card from a shuffled deck are examples.
- Outcome
- Each possible result of a random experiment. For example, getting 'head' when a coin is tossed, or getting 4 when a die is rolled.
- Equally likely outcomes
- Outcomes of an experiment that have the same chance of occurring. If there are n equally likely outcomes, each has probability 1/n.
- Event
- A collection of outcomes from an experiment that we are interested in. For instance, 'getting a number greater than 4' is an event when a die is rolled.
- Favourable outcomes
- The outcomes that belong to the given event. They are counted in the numerator of the probability formula.
- Sure event
- An event that must happen in every trial of the experiment. Its probability is 1. For example, getting a number from 1 to 6 when a die is rolled.
- Impossible event
- An event that can never occur when the experiment is performed. Its probability is 0. Example: getting 7 on an ordinary die.
- Complementary event
- For an event E, the complementary event is 'not E', meaning E does not occur. The two together cover all possible outcomes, so P(E) + P(not E) = 1.
- Sample space
- The set of all possible outcomes of a random experiment. When a coin is tossed the sample space is {Head, Tail}; when a die is rolled it is {1, 2, 3, 4, 5, 6}.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Theoretical probability | P(E) = m/n | m is the number of outcomes favourable to event E and n is the total number of equally likely outcomes. This applies when all outcomes are equally likely. |
| Complement of an event | P(not E) = 1 − P(E) | not E is the event that E does not occur. This is also written as P(E′) = 1 − P(E). |
| Range of probability | 0 ≤ P(E) ≤ 1 | Any probability must be a number between 0 and 1 inclusive. A probability may be written as a fraction, decimal or percentage, but the formula always gives a fraction. |
Practice questions with answers
1. What are the probabilities of a sure event and an impossible event?
A sure event always occurs, so its probability is 1. An impossible event never occurs, so its probability is 0. For example, when a die is rolled, getting a number from 1 to 6 is sure, while getting a 7 is impossible.
2. What is meant by complementary events? Give one example.
Two events are complementary if one occurs exactly when the other does not. Together they cover all possible outcomes and their probabilities add to 1. When a coin is tossed, 'head' and 'tail' are complementary events.
3. A coin is tossed once. Find the probability of getting a tail.
A fair coin has two equally likely outcomes, head and tail. The number of outcomes favourable to getting a tail is 1 out of a total of 2. Therefore P(tail) = 1/2.
4. A die is thrown once. Find the probability of getting an odd number.
The possible outcomes are 1, 2, 3, 4, 5 and 6, so there are 6 equally likely outcomes. The odd numbers are 1, 3 and 5, giving 3 favourable outcomes. Thus P(odd) = 3/6 = 1/2.
5. Two coins are tossed together. What is the probability of getting at least one tail?
The sample space is {HH, HT, TH, TT}, so there are 4 equally likely outcomes. At least one tail means HT, TH or TT, giving 3 favourable outcomes. Therefore P(at least one tail) = 3/4.
6. A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is a red queen.
A standard deck has 52 cards. The red queens are the queen of hearts and the queen of diamonds, so there are 2 favourable outcomes. Thus P(red queen) = 2/52 = 1/26.
7. A bag contains 6 white and 4 black marbles. One marble is drawn at random. What is the probability that the marble is white?
The total number of marbles is 6 + 4 = 10, and each is equally likely to be drawn. Since 6 marbles are white, P(white) = 6/10 = 3/5.
8. If the probability of an event E is 0.25, what is the probability that E does not happen?
The event that E does not happen is the complement of E. Since the sum of probabilities of an event and its complement is 1, we have P(not E) = 1 − 0.25 = 0.75.
9. Two dice are thrown together. Find the probability that both dice show the same number.
When two dice are thrown, there are 6 × 6 = 36 equally likely outcomes. The favourable outcomes are (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6), which are 6 outcomes. Hence the probability is 6/36 = 1/6.
10. One card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card is a black face card?
Black cards belong to clubs and spades, each suit having 13 cards. The face cards are king, queen and jack, so each black suit contributes 3 face cards, giving 3 × 2 = 6 black face cards. With 52 cards in total, P(black face card) = 6/52 = 3/26.
Studying Probability?
Turn your own textbook pages, a lecture recording or a YouTube video into notes like these — plus flashcards, quizzes and practice questions — in your own language. The Standard plan is free, permanently, and needs no card.
Make notes from your own material — free