Most Important Questions CBSE Class 10 Maths Probability with Solutions

Probability is a crucial concept in mathematics, both in the syllabus and in everyday life. From determining the possibility of rain tomorrow to calculating winning odds in games, probability is everywhere. In Class 10, probability is a topic that helps students sharpen their analytical thinking and problem-solving skills. Working through extra probability questions is a powerful way to understand this topic in detail and handle challenging exams.

Let us understand why Probability Important Questions Class 10 are essential, how they can benefit students, and effective strategies for using them to build a strong mathematical foundation.

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(Most Important Questions of this Chapter from our  book)

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Chapter 14 Probability: Important Questions

1. In a cards game, there are ten cards, 1 to 10. Two players, seated facing each othe randomly choose 5 cards each. They arrange their cards in ascending order of the number on the card as shown below.

The difference between the corresponding cards is calculated such that the lower value is subtracted from the higher value. In a random game, what is the probability that the sum of the differences is 24?

a. 0

b. ⅕

c. ½ 

d. (cannot be calculated without knowing the cards chosen by each player.)

Soln. a) 0

Given, two players choose randomly 5 cards and arrange them in increasing order.

For example: 

Player 1: 2,5,7,9,10

Player 2:1,3,4,6,8

Difference between them =1,2,3,3,2

Sum of  the differences =1+2+3+3+2

=12

As the sum of the difference is always > 24.

Probability =0. 

2. At a party, there is one last pizza slice and two people who want it. To decide who gets the last slice, two fair six-sided dice are rolled. If the largest number in the roll is:

1, 3 or 6, Ananya would get the last slice, and

2, 4 or 5, Pranit would get it.

In a random roll of dice, who has a higher chance of getting the last pizza slice?(Note: If the number on both the dice is the same, then consider that number as the larger number.)

a. Ananya

b. Pranit

c. Both have an equal chance

d. (cannot be answered without knowing the exact numbers in a roll.)

Soln. 

Since, the two fair six-sided dice are rolled.

The sample space of getting largest number in the roll is given as

S={1,1, 1,2, 1,3, 1,4, 1,5, 1,6, 2,2, 2,3, 2,4, 2,5, 2,6, 3,3, 3,4, 3,5, 3,6, 4,4, 4,5, 4,6, 5,5, 5,6, 6,6}

Let A be the event of getting the largest number in the roll of Ananya's number.

A=1,1, 1,3, 2,3, 3,3, 1,6, 2,6, 3,6, 4,6, 5,6, 6,6

⇒ n(A)=10

Let B be the event of getting the largest number in the roll of Pranit's number.

B={1,2, 2,2, 1,4, 2,4, 3,4, 4,4, 1,5, 2,5, 3,5, 4,5, 5,5}

⇒ n(B)=11

We know that,

Probablity of an event=number of elements in an event/total number of elements in sample space

∵ n(A)<n(B)

So, the chances of event B is more.

Hence, Pranit has a higher chance of getting the last pizza slice.

3. In a medical center, 780 randomly selected people were observed to find if there is a relationship between age and the likelihood of getting a heart attack. The following results were observed.

(i) Based on this table, what is the probability that a randomly chosen person from the same sample is younger than or equal to 55 years and has had a heart attack?

(ii) Looking at the data in the table, Giri says, "if a person is randomly chosen, then the probability that the person has had a heart attack is about 12.5%".

Is the statement true or false? Justify your reason.

Soln. 

(i) Probability that a randomly chosen person from the same sample is younger than or equal to 55 years and has had a heart attack

Soln. P(Younger than or equal to 55 and had a heart attack)=(Number of people younger than or equal to 55 who had a heart attack​/Total number of people)

P = 29/780

P ≈ 0.0372(or 3.72%).

(ii) Giri says, "if a person is randomly chosen, then the probability that the person has had a heart attack is about 12.5%". Is the statement true or false?

Soln. P(People who had a heart attack)=(People who had a heart attack​/Total number of people)

P = 104/780

P ≈ 0.133(or 13.75%).

Since 13.33%13.33\%13.33% is not approximately 12.5%12.5\%12.5%, Giri's statement is false.‍

4. Shown below is a square dart board with circular rings inside.

(Note: The figure is not to scale.)

Find the probability that a dart thrown at random lands on the shaded area. Show your steps.

Soln.

The diameter of the circle =4 units.

Then, radius of the circle=2 units

We know that, area of the circle =πr²

⇒ π×2²

=4π sq. units

Now, area of square =side²

⇒4²

=16 sq. units

Then the area of the shaded region = Area of square-Area of circle

=16-4π sq. units

Thus, the probability of hitting the shaded region when a dart is thrown randomly =16-4π

4=4-π.

Hence, the required probability is 4-π.

5. On a particular day, Vidhi and Unnati couldn't decide on who would get to drive the car. They had one coin each and flipped their coins exactly three times. The following was agreed upon:

If Vidhi gets two heads in a row, she would drive the car.

If Unnati gets a head immediately followed by a tail, she will drive the car.

Who has more probability of driving the car that day? List all outcomes and show your steps.

Soln.

Let S be the sample space

Since, one coin is tossed three times.

S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

So, n(S)=8

Let A be an event that the Vidhi drive the car.

Vidhi drive the car only if she gets heads in a row.

∴ A=HHH,HHT,THH

⇒n(A)=3

∴P(A)=n(A)/n(S)=3/8            ...1

Let B be an event that the Unnati drive the car.

Unnati drive the car only if she gets a head immediately followed by a tail.

∴ B=HHT, HTH, THT, HTT

⇒n(B)=4

∴P(B)=n(B)n(S)=4/8=1/2        ...2

Using equations 1 and 2,

PA>PB

If they flipped their coins exactly three times, Unnati is more likely to drive the car that day.

6. A 4-sided fair die is numbered 1 - 4. Nikhil and Pratik are playing with each other with such a die. They roll their dice once at the same time. A player wins only if they get a number larger than the other player.

What is the probability of Pratik winning the game? Show your work.

Let S be the sample space

Let N₁={1, 2, 3, 4} and N₂={1, 2, 3, 4}.

S=N1×N2={(x,y)|x∈N₁, y∈N₂}

So, n(S)=4×4=16

Let N₁ represent the outcomes of a 4-sided fair die by Nikhil and N₂ represent the outcomes of a 4-sides fair die by Pratik.

Let A be an event that the Pratik win the game.

∴ A=x,y|x<y, x∈N₁, y∈N₂

A={ (1,2), (1,3), (1,4), (2,3), (2,4), 3,4}

⇒n(A)=6

∴P(A)=n(A)/n(S)=6/16=3/8

Hence, the probability of Pratik winning the game is 3/8.

7. Shown below are two baskets with grey and black balls.

Abhishek is playing a game with his friend, and he has to close his eyes and pick a black ball from one of the baskets in one trial.

He said, "I will try with basket 2 as it has a higher number of black balls than basket 1, and hence the probability of picking a black ball from basket 2 is higher."

Is Abhishek's statement correct? Justify your answer.

Soln. 

No, Abhishek's statement is not correct.

Total balls in basket 1=3 Grey balls+4 Black balls

Number of possible outcomes in basket 1=3+4=7

Number of black balls in basket 1=4

Hence, the number of favourable outcomes=4

We have, PE=Number of favourable outcomes/Number of possible outcomes

⇒Pblack ball from basket 1=4/7      ...1

Now, total balls in basket 2=6 Grey balls+8 Black balls

Number of possible outcomes in basket 2=6+8=14

Number of black balls in basket 2=8

Hence, the number of favourable outcomes=8

We have, PE=Number of favourable outcomes/Number of possible outcomes

⇒Pblack ball from basket 2=814=2/7      ...2

From equations 1 and 2,

The probability of picking a black ball from basket 1 is same as basket 2.

8. Rohan has a bag of multiple balls either pink, green or yellow in colour. He randomly picks up one ball.

His friend, Farid, predicted, "The probability of Rohan picking a pink ball is definitely ⅓  as there are 3 colours".Is Farid's statement true or false? Give a valid reason or a counter-example.

Soln. 

Farid's statement is false.

We don't know the exact number of pink, green and yellow colour balls in a bag.

Let us consider, the bag has 1 pink ball, 2 green balls and 2 yellow balls.

Total balls in a bag=1 Pink balls+2 Green balls+2 Yellow balls

Number of possible outcomes=1+2+2=5

Number of pink balls in a bag=1

Hence, the number of favourable outcomes=1

We have, PE=Number of favourable outcomes/Number of possible outcomes

⇒pink ball=1/5      ...1

Hence, the probability of Rohan picking a pink ball is 1/5 not 1/3.

9. Shivesh was tossing a fair coin. Shown below are the outcomes of his first 5 tosses. [Tail Tail Tail Tail Tail ]

Is the probability of Shivesh getting a head in his sixth toss higher than the probability of getting a tail? Give a valid reason.

Soln. 

No, the probability of Shivesh getting a head in his sixth toss higher than the probability of getting a tail.

We know that,

If a fair coin is tossed the probability of getting a head is equal to the probability of getting a tail.

For example:

If a coin is tossed.

The possible outcomes are H, T

Probability of an event=Number of favourable outcomes/Number of possible outcomes

Hence, P(Head)=1/2=P(Tail)

10. At a fair, there is a game such that it has two bags. Bag 1 has an equal number of red(R) and yellow(Y) cubes and bag 2 has an equal number of red (R) and blue(B) cubes. Rohit has to pick a cube from each of the bags. If he picks up at least 1 red cube, he gets a prize.

Find the probability of Rohit getting a prize. List all outcomes and show your work.

Soln. 

Since, bag 1 has an equal number of red(R) and yellow(Y) cubes.

So, the probability of selecting a red(R) and yellow(Y) cube from the bag 1 are same.

Similarly, bage 2 has an equal number of red(R) and blue(B) cubes.

So, the probability of selecting a red(R) and blue(B) cube from the bage 2 are same.

Let S be the sample space of selecting a cube from each of the bags.

∴ S=RR, RB, YR, YB

⇒ n(S)=4

It is given that, if Rohit picks up atleast one red(R) cube, he gets a prize.

Let A be the event of Rohit getting a prize.

∴ A=RR, RB, YR

⇒ n(A)=3

We know that,

PE=Number of favourable outcomes/Number of possible outcomes

P(A)=¾ 

Hence, the probability of Rohit getting a prize is ¾ 

The Role of Probability in Mathematics 

Probability, often defined as the measure of the possibility of an event occurring, plays an important role in both theoretical and applied mathematics. This field of study is foundational to a range of subjects like statistics, finance, science, and engineering. In Class 10, understanding probability goes beyond learning definitions and formulas; it sets the stage for data interpretation and critical decision-making skills.

In mathematics, probability encourages logical reasoning and a structured approach to problem-solving. Students develop the ability to predict and solve Ch 14 Maths Class 10 Extra Questions by exploring the concept of probability. This skill is not only useful in exams but also in real-life situations, where decisions often hinge on understanding risks and probabilities.

Why Extra Questions Are Necessary for Probability Mastery

Standard textbook questions provide an introduction to probability, but extra questions are where real understanding begins. Extra questions on probability for CBSE Class 10 often involve complex scenarios and multi-step problems, which necessitate a deeper understanding of the subject. These extra questions serve several purposes:

• They introduce students to various types of probability questions, helping them apply different strategies and formulas.
• Extra questions push students beyond typical textbook problems, allowing them to approach challenging concepts with confidence and fluency.
• Exam questions can vary in difficulty, and handling extra questions from Probability Class 10 provides students with a sense of what to expect. It’s a form of pre-exam practice that strengthens time management and analytical thinking.
• Extra questions often involve real-life scenarios or abstract setups that require students to visualize, interpret, and calculate probabilities from scratch. This process builds the foundation for more advanced studies in mathematics and data science.

Core Concepts in Probability for Class 10

Before diving into extra questions, it’s essential to have a solid understanding of the core concepts in Class 10 Probability. These concepts serve as the building blocks for handling complex problems:

• In probability, a random experiment is an action or process that leads to uncertain results. Examples include rolling a die or drawing a card from a deck. Recognizing different types of experiments is crucial in solving probability questions.
• The sample space is the set of all possible outcomes in a random experiment. For example, the sample space for flipping a coin is {Heads, Tails}. Knowing how to define the sample space is important in probability calculations.
• An event is a specific outcome or a set of outcomes within the sample space. For example, getting an even number when rolling a die is an event.
• The following formula calculates the probability of an event happening:

P(E) = (Number of favourable outcomes)/ Total number of outcomes

How to Approach Probability Class 10 Extra Questions

Extra questions can seem intimidating, but a strategic approach can make them manageable and even enjoyable. Here are some practical tips for handling probability extra questions effectively:

• In probability, visualizing the scenario can be very helpful. If the problem involves rolling dice, drawing cards, or selecting colored balls, sketching the setup can provide a clearer picture of possible outcomes and events. For example, if a question involves selecting cards from a deck, it can help to think of the suits, ranks, and specific counts.
• Probability-extra questions often require multiple steps to arrive at a solution.  For example, calculating the probability of drawing two specific cards in succession may involve conditional probabilities or combinations. Divide the problem into smaller parts and solve each step individually to avoid feeling overwhelmed.
• Understanding whether an event is independent or dependent is crucial. Independent events are unaffected by previous outcomes, such as flipping a coin multiple times. Conversely, previous results influence dependent events, such as drawing cards without replacement. Recognizing these distinctions allows students to apply the right probability formulas.

Practice Different Formulas and Scenarios

In probability, various formulas apply to different situations, such as:

• The Addition Rule is useful when calculating the probability of one event or another occurring.
• The Multiplication Rule is applied when calculating the probability of two independent events occurring in sequence.
• The Complementary Rule applies when determining the likelihood of an event not occurring.

These formulas are often incorporated in probability extra questions, which require students to select the appropriate formula based on the specific context.

Strategies to Maximise Learning with Probability Important Questions for Class 10

Students should use effective study techniques that promote active learning to make the most of the important questions in Probability Class 10. Here are some recommended approaches:

• Active recall involves trying to answer questions without immediately looking at notes, which strengthens memory. When combined with spaced repetition, which involves revisiting questions at different intervals, this technique guarantees students' long-term retention of probability concepts.
• Discussing probability questions with classmates can reveal different perspectives and problem-solving methods. Group study can be particularly beneficial with complex probability questions, as peers may explain concepts in ways that resonate better than textbook explanations.
• Since exams are timed, practicing probability questions under time constraints can improve both speed and accuracy. Start with simpler problems and gradually increase the difficulty level to build confidence in tackling probability questions quickly and correctly.

Mastering probability is about more than just getting excellent grades; it’s about developing a logical and analytical mindset that will be useful throughout life. Extra questions from Probability Class 10 are crucial resources for enhancing problem-solving abilities, fostering self-assurance, and applying concepts to practical scenarios. Whether preparing for exams or exploring mathematics further, extra questions are a powerful resource that enhances understanding and enjoyment of this fascinating subject.

With consistent practice, clear visualization, and strategic approaches, students can build a strong foundation in probability and tackle even the most challenging problems with ease. Embrace these extra questions as opportunities to push beyond the basics and unlock the full potential of probability in both academic and real-life scenarios.