Class 10 — Mathematics (NCERT)
Probability (Theoretical Probability)
Class 10
- ✓By the end of this lesson students will be able to define probability and distinguish between experimental and theoretical probability.
- ✓By the end of this lesson students will be able to understand the terms: experiment, trial, event, elementary event, sample space, and favourable outcomes.
- ✓By the end of this lesson students will be able to calculate the theoretical probability of an event using the formula P(E) = (Number of favourable outcomes) / (Total number of possible outcomes).
- ✓By the end of this lesson students will be able to understand the concepts of sure event, impossible event, and complementary events.
- ✓By the end of this lesson students will be able to solve problems involving the probability of various events.
Key concepts
Probability is a measure of the likelihood of an event occurring. In Class IX, we studied experimental probability, which is based on actual experiments. In Class X, we will study theoretical probability (also known as classical probability), which is based on a priori reasoning without actually performing the experiment.
An operation which can produce some well-defined outcomes. Examples include tossing a coin, rolling a die, or drawing a card from a deck.
A single performance of an experiment. For example, each toss of a coin or each roll of a die is a trial.
A possible result of a trial. When a coin is tossed, 'Head' or 'Tail' are the possible outcomes.
The set of all possible outcomes of an experiment. For tossing a coin, the sample space S = {H, T}. For rolling a die, S = {1, 2, 3, 4, 5, 6}.
A collection of some outcomes of the experiment. For example, in rolling a die, getting an even number is an event E = {2, 4, 6}.
An event having only one outcome of the experiment. In rolling a die, getting '1' is an elementary event. Getting '2' is another elementary event.
Outcomes are said to be equally likely if each outcome has the same chance of occurring. For example, when a fair coin is tossed, Head and Tail are equally likely outcomes.
If all outcomes of an experiment are equally likely, then the theoretical probability of an event E is defined as the ratio of the number of outcomes favourable to E to the total number of possible outcomes.
An event that is certain to occur. The probability of a sure event is 1. For example, getting a number less than 7 when a die is rolled is a sure event.
An event that cannot occur. The probability of an impossible event is 0. For example, getting an 8 when a die is rolled is an impossible event.
For an event E, the event 'not E' (meaning E does not occur) is called the complementary event of E, denoted by Ē or E'. The sum of the probabilities of an event and its complementary event is always 1.
Key facts to remember
- 1The probability of an event E, denoted by P(E), always satisfies 0 ≤ P(E) ≤ 1.
- 2The sum of the probabilities of all the elementary events of an experiment is 1.
- 3An event which is impossible to occur has probability 0. Such an event is called an impossible event.
- 4An event which is certain to occur has probability 1. Such an event is called a sure event or a certain event.
- 5For any event E, P(E) + P(Ē) = 1, where Ē represents 'not E'. Ē is called the complementary event of E.
- 6Theoretical probability is based on the assumption that all outcomes of an experiment are equally likely.
Worked examples
Example 1
A coin is tossed once. What is the probability of getting a Head?
Answer
1/2
This is a basic example to illustrate the application of the theoretical probability formula.
Example 2
A die is thrown once. Find the probability of getting:\n(i) a prime number\n(ii) a number lying between 2 and 6\n(iii) an odd number
Answer
(i) 1/2 (ii) 1/2 (iii) 1/2
Remember to simplify the probability fraction to its lowest terms.
Example 3
A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is:\n(i) red?\n(ii) not red?
Answer
(i) 3/8 (ii) 5/8
This example demonstrates the concept of complementary events, which can often simplify calculations.
Common mistakes
- ✗Incorrectly identifying the total number of possible outcomes in the sample space.
- ✗Incorrectly identifying the number of favourable outcomes for a specific event.
- ✗Expressing probability as a value less than 0 or greater than 1, which is mathematically incorrect.
- ✗Confusing terms like 'at least', 'at most', 'exactly' when determining favourable outcomes.
- ✗Not simplifying the fraction representing the probability to its lowest terms.
Exam tips
- ★Always write down the formula for probability, P(E) = (Number of favourable outcomes) / (Total number of possible outcomes), before substituting values.
- ★Clearly list all possible outcomes (sample space) and favourable outcomes to avoid errors, especially in complex problems.
- ★Ensure your final answer for probability is always a fraction between 0 and 1 (inclusive), or a decimal/percentage equivalent.
- ★Read the question carefully to understand what constitutes a 'favourable outcome' and to avoid misinterpretations.
- ★For problems involving 'not E', consider using the complementary event formula P(Ē) = 1 - P(E) if calculating P(E) is easier.
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