# NCERT Solutions for Class 10 Maths Chapter 15

NCERT Solutions for Class 10 Maths Chapter 15 Exercise 15.1 and Exercise 15.2 Probability in English and प्रायिकता की प्रश्नावली 15.1 और प्रश्नावली 15.2 in Hindi medium for CBSE, UP Board, MP Board 2019-20 as well as Schools following CBSE Board free PDF download. NCERT Books for all subjects along with Offline Apps of NCERT Textbook’s Solutions for class 10 all subjects.

 Class 10: Maths – गणित Chapter 15: Probability ## NCERT Solutions for Class 10 Maths Chapter 15

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### 10 Maths Exercise 15.1 Solutions

NCERT Solutions for Class 10 Maths Chapter 15 Exercise 15.1 Probability is given below. Other exercises like Exercise 15.2 or Hindi Medium Solutions are also available. See the solutions of other chapters of Class 10 Maths or go to Top of the page.        ### 10 Maths Exercise 15.2 Solutions

NCERT Solutions for Class 10 Maths Chapter 15 Exercise 15.2 Probability is given below. Other exercises like Exercise 15.1 or Hindi Medium Solutions are also available. See the solutions of other chapters of Class 10 Maths or go to Top of the page.   ### 10 गणित के प्रश्नावली 15.1 के हल

NCERT Solutions for Class 10 Maths Chapter 15 Exercise 15.1 Probability is given below. Other exercises like प्रश्नावली 15.2 or English Medium Solutions are also available. See the solutions of other chapters of Class 10 Maths or go to Top of the page.          ### 10 गणित के प्रश्नावली 15.2 के हल

NCERT Solutions for Class 10 Maths Chapter 15 Exercise 15.2 Probability is given below. Other exercises like प्रश्नावली 15.1 or English Medium Solutions are also available. See the solutions of other chapters of Class 10 Maths or go to Top of the page.   #### Previous Years Questions

##### One mark questions
1. A number is chosen at random from the numbers -3, -2,-1, 0, 1, 2, 3. What will be the probability that square of this number is less than or equal to 1? [CBSE 2017]
##### Three marks questions
1. Two different dice are thrown together. Find the probability that the numbers obtained
(i) have a sum less than7
(ii) have a product less than 16
(iii) is a doublet of odd numbers. [CBSE 2017]
##### Four marks questions
1. Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25? [CBSE 2017]

The measure of certainty of events in numerical values, under certain conditions, is provided by the branch of mathematics called ‘Theory of Probability. This theory has extensive use as one of the basic tools in statistics and wide range of applications in Science, engineering, biological science, medical, commerce, weather forecasting etc.

#### Term related to Probability

• Random Experiment: A random experiment is one in which the exact outcome cannot be predicted.
• Trial: Performing a random experiment is called a trial.
• Outcomes: The result of a random experiment is called an outcome.
• Sample space: The collection of all possible outcomes of a random experiment is called a sample space.
• Event: Any possible outcome or combination of outcomes of a random experiment is called an event.
• Equally likely events: Two or more events of a random experiment are said to be equally likely events if each one of them have an equal chance of occurrence.
• Probability of an event: The chance of occurrence of the event expressed quantitatively is known as the probability of an event and denoted by P(E).

Historical facts!

• The concept of probability was developed in a very strange manner. In 1654, a gambler by name Chevalier de Mere approached the well-known 17th century French philosopher and mathematician Blaise Pascal regarding certain dice problems. Pascal discussed them with another French mathematician Pierre de Fermat and they found solution to dice problems. This work was the beginning of probability theory.
• Probability theory has its actual origin in the 16th century when an Italian physician and mathematician J. Cardan wrote the first book on the subject ‘The book on Games of Chance’. Since its inception, the study of Statistics and probability has attracted the attention of great mathematicians (James Bernoulli (1654 – 1705), A. de Moivre (1667 – 1754) and Pierre Simon Laplace (1749 – 1827)).
• In 1812, Pierre Simon Laplace or Pierre de Laplace (1749 – 1827, France) proposed a mathematical system of inductive reasoning based on probability. He introduced many principles of probability, one among them is, ‘Probability is the ratio of the favoured events to the total possible events’.
• Statistician Karl Pearson (1857 – 1936) had tossed the coin 24000 times and he got 12012 heads. Then calculated experimental probability 12012/2400 = 0.5005.
• In the eighteenth century French De Buffon tossed a coin 4040 times and got 2048 heads. Then he calculated experimental probability 2048/4040 = 0.507.

#### Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

When we toss a coin, the possible outcomes are only two, head or tail, which are equally likely outcomes. Therefore, the result of an individual toss is completely unpredictable.

#### यदि P(E)=0.05 है, तो ‘E नहीं’ की प्रायिकता क्या है?

हम जानते हैं कि P(E)+P(नहीं E)=1
इसलिए 0.05+P(नहीं E)=1
⇒ P(नहीं E)=1-0.05=0.95

#### एक थैले में केवल नीबू की महक वाली मीठी गोलियां हैं। मालिनी बिना थैले में झांके उसमे से एक गोली निकलती है। इसकी क्या प्रायिकता है कि वह निकाली गई गोली संतरे की महक वाली है?

थैले में संतरे की महक वाली मीठी गोलियां नहीं हैं इसलिए संतरे की गोली निकलना एक असंभव घटना है।
P(संतरे की महक वाली गोली ) = 0

#### It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

Probability that two students are not having same birthday P (not E) = 0.992
Probability that two students are having same birthday P (E)
= 1 − P (not E)
= 1 − 0.992
= 0.008

#### A die is thrown once. Find the probability of getting a prime number.

The possible outcomes when a dice is thrown = {1, 2, 3, 4, 5, 6}
Number of possible outcomes of a dice = 6
Prime numbers on a dice are 2, 3, and 5.
Total prime numbers on a dice = 3
Probability of getting a prime number = 3/6 =1/2

#### एक पिग्गी बैंक (piggy bank) में, 50 पैसे के सौ सिक्के हैं, ₹ 1 के पचास सिक्के हैं, ₹ 2 के बीस सिक्के और ₹ 5 के दस सिक्के हैं। यदि पिग्गी बैंक को हिलाकर उल्टा करने पर कोई एक सिक्का गिरने के परिणाम सम्प्रायिक हैं, तो इसकी क्या प्रायिकता है कि वह गिरा हुआ सिक्का 50 पैसे का होगा?

50 पैसे के सिक्के = 100
₹ 1 के सिक्के = 50
₹ 2 के सिक्के = 20
₹ 5 के सिक्के = 10
कुल सिक्के = 100 + 50 + 20 + 10 = 180

P(50 पैसे के सिक्के ) = (50 पैसे के सिक्के )/(कुल सिक्के ) = 100/180 = 5/9

#### गोपी अपने जल – जीव कुंड के लिए एक दुकान से मछली खरीदती है। दुकानदार एक टंकी, जिसमे 5 नर मछली और 8 मादा मछली हैं, में से एक मछली यादृच्छया उसे देने के किये निकलती है । इसकी क्या प्रायिकता है कि निकाली गई मछली नर मछली है ?

नर मछली = 5
मादा मछली = 8
कुल मछली = 5 + 8 = 13
P(नर मछली ) = (नर मछली)/(कुल मछली) = 5/13

#### Five cards−−the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. What is the probability that the card is the queen?

Total number of cards = 5
Total number of queens = 1
P(getting a queen)

= (Number of favourable outcomes)/(Number of total possible outcomes)
=1/5

#### A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?

Total number of bulbs = 20
Total number of defective bulbs = 4
P(getting a defective bulb)

= (Number of favourable outcomes)/(Number of total possible outcomes)
= 4/20
= 1/5

#### A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a two-digit number.

Total number of discs = 90
Total number of two-digit numbers between 1 and 90 = 81
P (getting a two-digit number)

= 91/90
= 9/10

#### 144 बॉल पेनों के एक समूह में 20 बॉल पेन ख़राब हैं और शेष अच्छे हैं। आप वही पेन खरीदना चाहेंगे जो अच्छा हो, परन्तु खराब पेन आप खरीदना नहीं चाहेंगे। दुकानदार इन पेनों में से, यादृच्छया एक पेन निकलकर आपको देता है। इसकी क्या प्रायिकता है कि आप वह पेन खरीदेंगे?

कुल पेन = 144
कुल ख़राब पेन = 20
कुल अच्छे पेन = 144 – 20 = 122
P(आप वह पेन खरीदेंगे)

= ( कुल अच्छे पेन)/( कुल पेन)
= 122/144
= 61/72

#### एक थैले में 5 लाल गेंद और कुछ नीली गेंदे हैं यदि इस थैले में से नीली गेंद निकालने की प्रायिकता लाल गेंद निकालने की प्रायिकता की दुगुनी है, तो थैले में नीली गेंदों की संख्या ज्ञात कीजिए।

माना नीली गेंदों की कुल संख्या = x
लाल गेंदों की कुल संख्या = 5
कुल गेंदे = x + 5
P (लाल गेंद) = 5/(5 + x)
P (नीली गेंद) = x/(5 + x)
दिया है,
2(5/(5 + x)) = x/(5 + x)
⇒ 10(x + 5) = x^2 + 5x
⇒ x^2-5x-50=0
⇒ x^2 – 10x + 5x – 50 = 0
⇒ x(x – 10) + 5 (x – 10) = 0
⇒ (x – 10)(x + 5) = 0
⇒ x – 10 = 0 या x + 5 = 0
⇒ x = 10 या x = – 5
क्योंकि गेंदों की संख्या ऋणात्मक नहीं हो सकती, अतः नीली गेंदों की कुल संख्या 10 है ।