CBSE Class 10 Mathematics (2026–27)
Chapter 14: Probability
20 Important Questions and Answers
Q1. What is probability? Explain with an example.
Answer:
Probability is a measure of the likelihood of an event occurring. It tells us how likely or unlikely an event is to happen. The probability of an event is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. The formula is:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
For example, when a fair coin is tossed, there are two possible outcomes: Head and Tail. If we want the probability of getting a Head, there is one favourable outcome and two total outcomes. Therefore, the probability is 1/2. Probability values always lie between 0 and 1, where 0 means impossible and 1 means certain.
Q2. Define experimental probability.
Answer:
Experimental probability is the probability obtained from actual experiments or observations. It is based on the number of times an event occurs during repeated trials. The formula is:
Experimental Probability = Number of Times an Event Occurs / Total Number of Trials
For example, if a coin is tossed 100 times and Heads appear 55 times, then the experimental probability of getting a Head is 55/100 = 0.55. Experimental probability may differ from theoretical probability due to chance variations. However, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. It is widely used in real-life situations involving observations and data collection.
Q3. What is meant by an event in probability?
Answer:
An event is a particular outcome or a group of outcomes of an experiment. In probability, an event represents the result in which we are interested. For example, when a die is rolled, getting a number greater than 4 is an event. The favourable outcomes are 5 and 6. Therefore, the event consists of two outcomes. Events can be simple or compound. A simple event has only one outcome, such as getting a 3 on a die. A compound event contains more than one outcome. Understanding events helps us calculate probabilities accurately and solve practical problems involving chance and uncertainty.
Q4. Find the probability of getting an even number when a die is rolled.
Answer:
A standard die has six faces numbered 1 to 6. The possible outcomes are {1, 2, 3, 4, 5, 6}. The even numbers among these are 2, 4, and 6. Therefore, the favourable outcomes are three.
Using the probability formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 3/6
= 1/2
Thus, the probability of getting an even number when a die is rolled is 1/2. This means there is a 50% chance of obtaining an even number. Probability helps us predict the likelihood of outcomes in random experiments such as rolling dice or tossing coins.
Q5. What is the probability of drawing a king from a deck of cards?
Answer:
A standard deck contains 52 playing cards. Among these, there are 4 kings, one from each suit. Since each card has an equal chance of being selected, the probability of drawing a king can be calculated using the formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 4/52
= 1/13
Therefore, the probability of drawing a king from a well-shuffled deck is 1/13. This means that out of every 13 selections, one king is expected on average. Card-related probability problems are common examples of theoretical probability and help students understand the concept of equally likely outcomes.
Q6. Explain impossible and sure events.
Answer:
An impossible event is an event that cannot occur under any circumstances. Its probability is always 0. For example, getting a number 7 when rolling a standard die is impossible because a die has only six faces.
A sure event is an event that is certain to occur. Its probability is always 1. For example, getting a number less than 7 when rolling a standard die is a sure event because all outcomes are between 1 and 6.
These two concepts represent the extreme values of probability. Every probability lies between 0 and 1. Understanding impossible and sure events helps in interpreting probability values correctly.
Q7. What are equally likely outcomes?
Answer:
Equally likely outcomes are outcomes that have the same chance of occurring. In such situations, no outcome is favoured over another. For example, when a fair coin is tossed, Head and Tail are equally likely outcomes because each has a probability of 1/2.
Similarly, when a fair die is rolled, each number from 1 to 6 has an equal chance of appearing. Equally likely outcomes are important because the formula for theoretical probability assumes that all outcomes are equally likely. If outcomes are not equally likely, different methods may be needed to calculate probabilities. Many probability problems in mathematics are based on this principle.
Q8. Find the probability of getting a prime number on a die.
Answer:
When a die is rolled, the possible outcomes are 1, 2, 3, 4, 5, and 6. The prime numbers among these are 2, 3, and 5. Therefore, the number of favourable outcomes is 3.
Using the formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 3/6
= 1/2
Hence, the probability of getting a prime number on a die is 1/2. This means that there is an equal chance of getting a prime number or a non-prime number. Such questions help students understand how to identify favourable outcomes before applying the probability formula.
Q9. What is a random experiment?
Answer:
A random experiment is an experiment whose outcome cannot be predicted with certainty in advance, even though all possible outcomes are known. Examples include tossing a coin, rolling a die, or drawing a card from a deck. In a random experiment, each trial is independent and may produce different outcomes.
For example, when a coin is tossed, we know that either Head or Tail can occur, but we cannot predict the exact outcome beforehand. Random experiments form the foundation of probability. By studying repeated random experiments, mathematicians determine the likelihood of various events and develop probability models for real-life situations.
Q10. What is the probability of getting a red card from a deck?
Answer:
A standard deck of cards contains 52 cards. Out of these, 26 cards are red, consisting of Hearts and Diamonds. Therefore, the number of favourable outcomes is 26.
Using the probability formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 26/52
= 1/2
Thus, the probability of drawing a red card from a well-shuffled deck is 1/2. This indicates that there is a 50% chance of selecting a red card. Card-based probability problems are useful for understanding equally likely outcomes and applying probability concepts effectively.
Q11. Why does probability lie between 0 and 1?
Answer:
Probability always lies between 0 and 1 because it represents the likelihood of an event occurring. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. Since favourable outcomes cannot exceed the total number of outcomes, the ratio:
Probability = Favourable Outcomes / Total Outcomes
can never be less than 0 or greater than 1. For example, the probability of getting a Head on a fair coin is 1/2, which lies between 0 and 1. This range helps us compare the chances of different events and interpret probability values correctly in practical situations.
Q12. Find the probability of selecting a vowel from the word “MATHEMATICS”.
Answer:
The word “MATHEMATICS” contains 11 letters. The vowels present are A, E, A, I, making a total of 4 vowels.
Therefore:
Total letters = 11
Favourable outcomes = 4
Using the probability formula:
Probability = Favourable Outcomes / Total Outcomes
= 4/11
Hence, the probability of selecting a vowel at random from the letters of “MATHEMATICS” is 4/11. Such problems help students apply probability concepts to everyday situations involving letters, numbers, and objects. Careful counting of total and favourable outcomes is essential for obtaining the correct answer.
Q13. Distinguish between theoretical and experimental probability.
Answer:
Theoretical probability is calculated using mathematical reasoning and assumes that all outcomes are equally likely. It is found using the formula:
Probability = Favourable Outcomes / Total Outcomes
Experimental probability is based on actual observations and results obtained from repeated trials. It is calculated by dividing the number of times an event occurs by the total number of trials.
For example, the theoretical probability of getting a Head on a coin is 1/2. However, if a coin is tossed 100 times and Heads occur 48 times, the experimental probability becomes 48/100. As the number of trials increases, experimental probability approaches theoretical probability.
Q14. Find the probability of getting a number less than 5 on a die.
Answer:
A standard die has six outcomes: 1, 2, 3, 4, 5, and 6. Numbers less than 5 are 1, 2, 3, and 4. Therefore, there are four favourable outcomes.
Using the probability formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 4/6
= 2/3
Thus, the probability of getting a number less than 5 when a die is rolled is 2/3. This means that in most cases, a number less than 5 is more likely to occur than a number greater than or equal to 5. Probability helps quantify such likelihoods mathematically.
Q15. What is a sample space?
Answer:
A sample space is the set of all possible outcomes of a random experiment. It is usually denoted by S. Every probability problem begins with identifying the sample space correctly.
For example, when a coin is tossed, the sample space is {H, T}. When a die is rolled, the sample space is {1, 2, 3, 4, 5, 6}. The total number of outcomes in the sample space is used as the denominator in probability calculations. Understanding sample spaces helps students identify favourable outcomes and calculate probabilities accurately. It is one of the most important concepts in probability theory.
Q16. Find the probability of drawing an ace from a pack of cards.
Answer:
A standard deck contains 52 cards. There are 4 aces in the deck, one from each suit. Therefore, the number of favourable outcomes is 4.
Using the probability formula:
Probability = Number of Favourable Outcomes / Total Number of Outcomes
= 4/52
= 1/13
Hence, the probability of drawing an ace from a well-shuffled deck of cards is 1/13. This means that among every 13 cards selected on average, one is expected to be an ace. Such examples help students understand probability calculations involving card games and random selections.
Q17. What is the probability of getting Tail when a coin is tossed?
Answer:
When a fair coin is tossed, there are two equally likely outcomes: Head and Tail. Since we are interested in getting a Tail, there is one favourable outcome.
Total possible outcomes = 2
Favourable outcomes = 1
Applying the probability formula:
Probability = Favourable Outcomes / Total Outcomes
= 1/2
Therefore, the probability of getting a Tail is 1/2. This indicates a 50% chance of obtaining a Tail in a single toss. Coin-toss experiments are among the simplest examples used to introduce probability because the outcomes are equally likely and easy to understand.
Q18. Find the probability of selecting an odd number from 1 to 10.
Answer:
The numbers from 1 to 10 are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
The odd numbers are:
1, 3, 5, 7, 9
Thus:
Total outcomes = 10
Favourable outcomes = 5
Using the formula:
Probability = Favourable Outcomes / Total Outcomes
= 5/10
= 1/2
Hence, the probability of selecting an odd number from 1 to 10 is 1/2. This means that odd and even numbers are equally represented in the given set, making the chances of selecting either type the same.
Q19. What do you understand by favourable outcomes?
Answer:
Favourable outcomes are those outcomes that satisfy the condition of a given event. They are the outcomes we are interested in when calculating probability. For example, when a die is rolled and the event is “getting an even number,” the favourable outcomes are 2, 4, and 6.
The number of favourable outcomes forms the numerator of the probability formula. Correct identification of favourable outcomes is essential because an incorrect count leads to wrong probability values. In every probability problem, students must first determine the sample space and then identify which outcomes are favourable to the event under consideration.
Q20. Find the probability of getting a multiple of 3 on a die.
Answer:
A standard die has six possible outcomes: 1, 2, 3, 4, 5, and 6. The multiples of 3 among these numbers are 3 and 6.
Therefore:
Favourable outcomes = 2
Total outcomes = 6
Using the probability formula:
Probability = Favourable Outcomes / Total Outcomes
= 2/6
= 1/3
Hence, the probability of getting a multiple of 3 when a die is rolled is 1/3. This means that out of every three rolls on average, one roll is expected to result in a multiple of 3. Probability helps us estimate such chances mathematically and systematically.