NCERT Solutions for Class 9 Ganita Manjari Chapter 7 The Mathematics of Maybe: Introduction to Probability
Table of Contents
Exercise Set 7.1
Question 1.
Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking.
(i) The next Monday will come after Sunday.
(ii) It will snow in Mumbai in July.
(iii) An elephant will walk through your classroom today.
(iv) You will greet at least one friend at school tomorrow.
Solution:
(i) The next Monday will come after Sunday.
Rank: 1
Label: Certain
Reason: Monday always comes immediately after Sunday in the weekly calendar.
श्रेणी: निश्चित
कारण: सप्ताह के क्रम में रविवार के बाद हमेशा सोमवार आता है।
(ii) It will snow in Mumbai in July.
Rank: 0
Label: Impossible
Reason: Mumbai has a warm, tropical climate, and snowfall in July is not expected.
मुंबई की जलवायु गर्म और उष्णकटिबंधीय है। जुलाई में वहाँ बर्फबारी होना सामान्य परिस्थितियों में असंभव है।
(iii) An elephant will walk through your classroom today.
Rank: Less than 0.5
Label: less likely
Reason: It is possible in imagination, but extremely unlikely in a normal classroom.
सामान्य परिस्थितियों में कक्षा के अंदर हाथी के आने की संभावना बहुत कम है।
(iv) You will greet at least one friend at school tomorrow.
Rank: Close to 1 (e.g., 0.8 or 0.9)
Label: More likely
Reason: If you go to school tomorrow, it is quite likely that you will meet and greet at least one friend.
यदि आप कल स्कूल जाते हैं, तो किसी मित्र से मिलकर उसका अभिवादन करने की संभावना अधिक है।
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NCERT Solutions for Class 9 Ganita Manjari Chapter 7 The Mathematics of Maybe: Introduction to Probability
Exercise Set 7.2
Question 1.
A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour:
10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets
(i) Calculate the probability that a randomly picked sweet from the sample is green.
(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
Solution:
Number of all possible outcomes in sample = 10 + 8 + 7 + 5 = 30
(i) Number of favourable outcomes = 8 (green sweets)
P (picking a green sweet) = \(\frac{8}{30}\) = 0.266… = 0.267 or 26.7%
(ii) Number of favourable outcomes = 7 (yellow sweets in sample)
P(picking a yellow sweet from the sample) = \(\frac{7}{30}\) = 0.2333… or 23.3%.
Estimated yellow sweets in 600 = \(\frac{7}{30} \times 600\) = 140
Question 2.
A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are:
14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club
Assume there are 800 students in the whole school.
(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
Solution:
Number of all possible outcomes in sample = 14 + 11 + 9 + 6 = 40
(i) Number of favourable outcomes = 11 (students preferring the Arts club)
P (A student prefers Arts Club) = \(\frac{11}{40}\) = 0.275 or 27.5%
(ii) Number of favourable outcomes = 9 (students preferring the Sports Club)
P (A student prefers Sports Club) = \(\frac{9}{40}\)
Estimate for 800 students = \(\frac{9}{40}\) × 800 = 180 students likely to prefer sports club.
Question 3.
Toss a coin 20 times and record the result each time (heads or tails).
(i) How many times did you get heads?
(ii) How many times did you get tails?
(iii) Calculate the experimental probability of getting heads.
(iv) If you toss the coin once more, what is the probability of getting tails?
Solution:
Do it yourself.
Question 4.
Toss a paper cup into the air 100 times. After each toss, record whether the cup lands on its bottom, upside down on its top or on its side (See Fig. 7.5). Assign probabilities to the outcomes by using experimental probability.
Solution:
Do it yourself.
Question 5.
What is the probability of getting an even number when rolling a fair 6-sided die?
Solution:
Number of all possible outcomes = 6 (numbers 1, 2, 3, 4, 5, 6)
Number of favourable outcomes = 3 (even numbers 2, 4, 6)
P(getting an even number) = Number of favourable outcomes / Number of all possible outcomes
= 3/6 = 1/2
= 0.5 or 50%
Question 6.
Suppose you roll a 6-sided die 12 times and get a ‘3’ three times.
(i) What is the experimental probability of rolling a ‘3’?
(ii) What is the theoretical probability of rolling a ‘3’?
(iii) Why might these probabilities be different? What would you expect to happen if you rolled the die 60, 600, or 6000 times?
Solution:
(i) Number of all outcomes = 12 (Total rolls in experiment)
Number of favourable outcomes = 3 (times ‘3’ actually appeared)
P(rolling a 3) = Number of favourable outcomes Number of all outcomes
= 3/12
= 0.25 or 25%
(ii) Number of all possible outcomes = 6 (numbers 1 through 6)
Number of favourable outcomes = 1 (only the number 3)
P(rolling a 3) = Number of favourable outcomes Number of all outcomes
= 1/6
= 0.167 or 16.7%
(iii) Difference between Experimental and Theoretical probabilities: The difference exists because experimental probability is based on evidence from a limited sample, while theoretical probability is based on the ideal mathematical outcome. As the number of trials increases, the experimental probability will likely get closer to the theoretical one. This is why a larger sample size makes our estimates more reliable.
