What does 'randomness' refer to in probability?

A situation where you cannot predict exactly what will happen.

Predicting the exact outcome of an event.

A situation where you cannot predict exactly what will happen.

An event with only one possible outcome.

The certainty of an event occurring.

Which of these events has an 'even chance' of occurring?

Flipping a coin and getting heads.

Drawing a red card from a deck of 52 cards.

Picking a blue marble from a bag with 10 blue and 1 red marble.

Which of the following describes an event that is 'less likely'?

Flipping a coin and getting heads.

Drawing a black card from a deck of 52 cards.

Assertion (A): The probability of an event cannot be negative. Reason (R): The number of favorable outcomes and the total number of outcomes are always non-negative.

A and R both correct, R explains A

Is this Class 9 Maths Chapter 7 Probability quiz free?

Yes, absolutely! All quizzes on Shrutam, including this one for Class 9 Maths Chapter 7, are completely free and require no signup.

How many questions are in this quiz?

This quiz contains 40 questions in Game Mode, covering various difficulty levels from easy to hard, plus additional questions in Book Questions and Monthly Test sections.

Is this quiz based on the NCERT syllabus?

Yes, all questions are meticulously aligned with the NCERT Class 9 Maths syllabus for Chapter 7, 'The Mathematics of Maybe: Introduction to Probability', ensuring comprehensive coverage of the topics.

What topics does this Probability quiz cover?

This quiz covers key concepts such as the definition of probability, the probability scale, experimental probability, theoretical probability, sample spaces, events, and the use of tree diagrams.

Can I use this quiz to prepare for my school exams?

Definitely! This quiz is an excellent resource for self-assessment and exam preparation, helping you identify your strengths and weaknesses in the chapter on Probability.

Are there explanations for the answers?

Yes, every question comes with a short explanation for quick feedback and a detailed explanation to help you understand the reasoning behind the correct answer.

Home › CBSE › Class 9 › Maths › The Mathematics of Maybe: Introduction to Probability

CBSE · Class 9 · 🧮 Maths · Chapter 7

The Mathematics of Maybe: Introduction to Probability

Probability scaleRandom eventsExperimental probabilityTheoretical probabilitySample spaceEvents

Chapter 7, 'The Mathematics of Maybe: Introduction to Probability', introduces students to the fundamental concepts of probability. It covers what probability is, how it's measured on a scale from 0 to 1, and the distinction between experimental and theoretical probability. Students will learn about sample spaces, events, and how to use tree diagrams to visualize outcomes. Understanding probability is crucial for developing logical reasoning and is applied in various real-world scenarios, from weather forecasting to financial analysis.

Probability Kya Hai? (What is Probability?)

Probability ek branch hai mathematics ki jo random events ke hone ki likelihood ko quantify karti hai. Iska matlab hai ki hum kitne sure hain ki koi event hoga ya nahi.

Random Event: Ek event jiska outcome pehle se predict nahi kiya ja sakta, jaise coin toss ya dice roll. Outcomes pata hote hain, par kaunsa aayega, yeh nahi pata.
Likelihood: Kisi event ke hone ki possibility. Isko words mein express kar sakte hain (impossible, less likely, equally likely, more likely, certain) ya numbers mein (0 se 1 tak).

Probability Scale

Probability ko ek scale par measure kiya jaata hai jo 0 se 1 tak hota hai:

0 (Impossible Event): Jis event ke hone ki koi chance nahi hai. Jaise, "Suraj ka West se nikalna".
1 (Certain Event): Jis event ka hona definite hai. Jaise, "Suraj ka East se nikalna".
0 aur 1 ke beech: Most events ki probability 0 aur 1 ke beech mein hoti hai, jo unki likelihood batati hai.
0.5 (Equally Likely): Jab do outcomes ke hone ke chances barabar hon. Jaise, fair coin toss mein Head ya Tail aana.
0.5 se kam (Less Likely): Event ke hone ke chances kam hain.
0.5 se zyada (More Likely): Event ke hone ke chances zyada hain.

Example: | Event | Likelihood (Words) | Probability (Number) | |-------------------------------------------|--------------------|----------------------| | Die par 7 aana | Impossible | 0 | | Coin toss mein Head aana | Equally Likely | 0.5 | | Bag mein sirf Red balls hon aur Red ball nikalna | Certain | 1 |

📖Definition

Probability: Kisi event ke hone ki numerical measure. Hamesha 0 aur 1 ke beech hoti hai.

❗Important

Probability kabhi bhi negative nahi ho sakti aur na hi 1 se zyada.

Probability ko Kaise Measure Karein? (Measuring Probability)

Probability ko measure karne ke do main tareeke hain:

Experimental Probability (Empirical Probability): Observations ya experiments perform karke.
Theoretical Probability (Classical Probability): Reasoning aur logic ka use karke.

Experimental Probability

Yeh probability actual experiments perform karke determine ki jaati hai. Jitni baar experiment kiya jaata hai (trials), utni hi accurate probability milti hai.

Formula:

$$P(E) = \frac{\text{Number of times the event occurred}}{\text{Total number of trials}}$$

Example: Ek coin ko 100 baar toss kiya. Heads 48 times aaye aur Tails 52 times aaye.
P(Heads) = $\frac{48}{100} = 0.48$
P(Tails) = $\frac{52}{100} = 0.52$

Theoretical Probability

Yeh probability bina experiment kiye, logic aur reasoning ke base par calculate ki jaati hai. Ismein assume kiya jaata hai ki saare possible outcomes equally likely hain.

Formula:

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

Favourable Outcome: Woh outcome jo hum chahte hain ki ho.
Total Possible Outcomes: Ek experiment ke saare possible results.

Example: Ek fair die roll kiya.
Total possible outcomes = $\{1, 2, 3, 4, 5, 6\}$. So, Total outcomes = 6.
Event E: "Getting an even number" = $\{2, 4, 6\}$. So, Favourable outcomes = 3.
P(Even number) = $\frac{3}{6} = \frac{1}{2} = 0.5$

Experimental vs. Theoretical Probability

| Feature | Experimental Probability | Theoretical Probability | |-------------------|---------------------------------------------------------|--------------------------------------------------------| | Basis | Actual observations/experiments | Logic, reasoning, equally likely outcomes assumption | | Calculation | Number of times event occurred / Total trials | Number of favourable outcomes / Total possible outcomes| | Accuracy | More trials = More accurate | Assumes ideal conditions | | Use Case | Real-world scenarios, historical data, simulations | Idealized situations, games of chance |

📖Definition

Experimental Probability: Jitni baar event hua / Total trials. Observation-based.

📖Definition

Theoretical Probability: Favourable outcomes / Total possible outcomes. Logic-based.

💡Tip

CBSE Class 9 mein mostly Theoretical Probability par questions aate hain, jab tak explicitly experimental na bola ho.

Probability ke Elements: Sample Spaces aur Events (Elements of Probability: Sample Spaces and Events)

Probability calculate karne ke liye, sabse pehle humein experiment ke saare possible outcomes ko samajhna zaroori hai.

Sample Space (S)

Definition: Ek random experiment ke saare possible outcomes ka set. Isse $S$ se denote karte hain.
Sample Size (n(S)): Sample space mein total number of outcomes. Isse $n(S)$ se denote karte hain.

Examples:

Coin Toss: $S = \{\text{Head, Tail}\}$, $n(S) = 2$
Rolling a Die: $S = \{\text{1, 2, 3, 4, 5, 6}\}$, $n(S) = 6$
Tossing two coins: $S = \{\text{HH, HT, TH, TT}\}$, $n(S) = 4$
Drawing a card from a deck: $S = \{\text{All 52 cards}\}$, $n(S) = 52$

Event (E)

Definition: Sample space ka koi bhi subset. Yeh ek single outcome ho sakta hai ya multiple outcomes ka collection.
Favourable Outcomes (n(E)): Event $E$ mein jitne outcomes hain, unki sankhya. Isse $n(E)$ se denote karte hain.

Examples (using previous sample spaces):

Coin Toss:
Event E1: "Getting a Head" $E_1 = \{\text{Head}\}$, $n(E_1) = 1$
Rolling a Die:
Event E2: "Getting an even number" $E_2 = \{\text{2, 4, 6}\}$, $n(E_2) = 3$
Event E3: "Getting a number greater than 4" $E_3 = \{\text{5, 6}\}$, $n(E_3) = 2$
Tossing two coins:
Event E4: "Getting at least one Head" $E_4 = \{\text{HH, HT, TH}\}$, $n(E_4) = 3$

Probability of an Event (P(E))

Once humein $n(S)$ aur $n(E)$ pata chal jaate hain, theoretical probability calculate karna easy ho jaata hai:

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)}$$

Important Properties:

$0 \le P(E) \le 1$ (Probability hamesha 0 aur 1 ke beech hoti hai)
$P(\text{Impossible Event}) = 0$
$P(\text{Certain Event}) = 1$
Complementary Events: Agar $E$ ek event hai, toh $E'$ (ya $\bar{E}$) "not E" ko denote karta hai. Yaani, event $E$ ka na hona.
$P(E) + P(E') = 1$
Iska matlab hai $P(E') = 1 - P(E)$
Example: Agar P(rain) = 0.3, toh P(not rain) = $1 - 0.3 = 0.7$.

📖Definition

Sample Space (S): All possible outcomes of an experiment. $n(S)$ is the count.

📖Definition

Event (E): Sample space ka subset. $n(E)$ is the count of favourable outcomes.

🧮Formula

$$P(E) = \frac{n(E)}{n(S)}$$ $$P(E') = 1 - P(E)$$

Tree Diagrams

Tree diagrams ek visual representation hain jo multi-step experiments ke saare possible outcomes ko list karne mein help karte hain.

Multi-step Experiment

Ek experiment jismein ek se zyada trials ya steps hote hain. Jaise, do coins toss karna, ya ek coin toss karke phir die roll karna.

Tree Diagram Kaise Banayein

Start Node: Ek starting point se shuru karo.
Branches for First Step: Pehle step ke har possible outcome ke liye ek branch nikalo. Har branch par outcome ko label karo.
Branches for Subsequent Steps: Har first-step outcome ke end se, second step ke har possible outcome ke liye branches nikalo. Isse repeat karte raho jab tak saare steps complete na ho jaayein.
List Outcomes: Har path (start se end tak) ek unique outcome represent karta hai. Saare end points par outcomes ko list karo.

Example: Tossing two coins

Step 1: First Coin Toss
Outcome 1: Head (H)
Outcome 2: Tail (T)
Step 2: Second Coin Toss (har first outcome ke baad)
Agar First H aaya: Second coin H ya T ho sakta hai.
Agar First T aaya: Second coin H ya T ho sakta hai.

[IMAGE: TODO: Tree diagram for two coin tosses showing HH, HT, TH, TT]

Outcomes from Tree Diagram:

So, Sample Space $S = \{\text{HH, HT, TH, TT}\}$ aur $n(S) = 4$.

Example: Rolling a die and then tossing a coin

Step 1: Die Roll (Outcomes: 1, 2, 3, 4, 5, 6)
Step 2: Coin Toss (Outcomes: H, T) - har die outcome ke baad

[IMAGE: TODO: Tree diagram for rolling a die and tossing a coin]

Outcomes from Tree Diagram:

1H, 1T
2H, 2T
3H, 3T
4H, 4T
5H, 5T
6H, 6T

So, Sample Space $S = \{\text{1H, 1T, 2H, 2T, ..., 6H, 6T}\}$ aur $n(S) = 12$.

Tree diagrams complex experiments ke liye sample space ko visualize aur list karne mein bohot helpful hote hain.

❗Important

Tree diagrams multi-step experiments ke liye useful hote hain, jahan outcomes ko systematically list karna hota hai.

Easy (15)

What is the range of values for probability?
- 0 to 1 (correct)
- -1 to 1
- 0 to 100
- Any real number
Why: Probability is measured on a scale from 0 to 1, inclusive.
An event that is impossible has a probability of:
- 1
- 0 (correct)
- 0.5
- Cannot be determined
Why: An impossible event has a probability of 0.
What does 'randomness' refer to in probability?
- Predicting the exact outcome of an event.
- A situation where you cannot predict exactly what will happen. (correct)
- An event with only one possible outcome.
- The certainty of an event occurring.
Why: Randomness means unpredictability of an exact outcome.
If an event is 'certain' to happen, its probability is:
- 0
- 0.5
- 1 (correct)
- Undefined
Why: A certain event has a probability of 1.
Which type of probability is determined by reasoning about the number of favourable outcomes in relation to the total number of possible outcomes?
- Experimental probability
- Subjective probability
- Theoretical probability (correct)
- Conditional probability
Why: Theoretical probability is based on reasoning about outcomes.
What is the term for the list of all possible outcomes of a random experiment?
- Event space
- Outcome set
- Sample space (correct)
- Probability set
Why: The sample space is the set of all possible outcomes.
Which of these events has an 'even chance' of occurring?
- Rolling a 6 on a standard die.
- Flipping a coin and getting heads. (correct)
- Drawing a red card from a deck of 52 cards.
- Picking a blue marble from a bag with 10 blue and 1 red marble.
Why: Flipping a coin has two equally likely outcomes: heads or tails.
What is the probability of picking a red sweet from a bag containing only red sweets?
- 0
- 0.5
- 1 (correct)
- Cannot be determined
Why: If all sweets are red, picking a red sweet is a certain event.
The probability of an event can be greater than 1.
Why: Probability values are always between 0 and 1, inclusive.
Experimental probability is calculated by performing observations or experiments.
Why: Experimental probability is based on actual trials and observed frequencies.
A tree diagram is used to list all possible outcomes of a multi-step experiment.
Why: Tree diagrams visually represent outcomes of sequential events.
The number of elements in the sample space is called the sample ______.
Why: The count of outcomes in a sample space is its size.
An event is a _______ of a sample space.
Why: An event consists of one or more outcomes from the sample space.
Probability is a type of measurement used to measure the _______ of events.
Why: Probability quantifies how likely an event is to occur.
The probability scale ranges from 0 to _______.
Why: The probability scale is a continuous range from 0 to 1.

Medium (15)

A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball?
- 3/8
- 5/8 (correct)
- 1/2
- 5/3
Why: Probability is (favorable outcomes) / (total outcomes).
When a fair die is rolled, what is the probability of getting an even number?
- 1/6
- 1/3
- 1/2 (correct)
- 2/3
Why: There are 3 even numbers (2, 4, 6) out of 6 total outcomes.
A coin is tossed 100 times, and heads appears 45 times. What is the experimental probability of getting tails?
- 0.45
- 0.5
- 0.55 (correct)
- 1
Why: Tails occurred 100 - 45 = 55 times out of 100 trials.
Which of the following describes an event that is 'less likely'?
- Flipping a coin and getting heads.
- Rolling a 3 on a standard die. (correct)
- The sun rising in the east.
- Drawing a black card from a deck of 52 cards.
Why: Rolling a 3 has a probability of 1/6, which is less than 1/2.
If a spinner has 4 equal sections colored red, blue, green, and yellow, what is the probability of landing on green or yellow?
- 1/4
- 1/2 (correct)
- 3/4
- 1
Why: There are 2 favorable outcomes (green, yellow) out of 4 total.
Match the terms with their definitions.
Why: These are the core definitions from the chapter.
Match the event with its likelihood.
Why: These match the examples given in the NCERT text.
Match the type of probability with its characteristic.
Why: These are the core distinctions between probability types and related concepts.
What is the probability of rolling a number less than 3 on a standard six-sided die?
Why: Numbers less than 3 are 1 and 2. So, 2 out of 6 outcomes.
If a spinner has 5 equal sections, what is the probability of landing on any specific section?
Why: Each section has an equal chance, so 1 out of 5.
Arrange the following events in increasing order of their likelihood: (A) Rolling a 7 on a standard die, (B) Getting heads on a coin toss, (C) The sun rising tomorrow, (D) Rolling an even number on a die.
Why: Probabilities are 0, 1/2, 1/2, 1. Order by value.
Arrange the steps to calculate experimental probability: (A) Count the number of times the event occurred, (B) Divide the number of event occurrences by the total number of trials, (C) Perform the experiment multiple times, (D) Identify the event of interest.
Why: First define, then perform, count, and finally calculate the ratio.
Refer to the image showing a standard die. What is the probability of rolling a number that is a prime number?
- 1/6
- 1/3
- 1/2 (correct)
- 2/3
Why: Prime numbers on a die are 2, 3, 5. So 3 out of 6 outcomes.
Consider the image of a coin. If this coin is tossed once, what is the probability of getting a 'Head'?
- 0
- 0.25
- 0.5 (correct)
- 1
Why: A fair coin has two equally likely outcomes: Head or Tail.
The image shows a probability scale. Which point on the scale represents an event that is 'more likely' than an even chance but not certain?
- Point A (near 0)
- Point B (at 0.5)
- Point C (between 0.5 and 1) (correct)
- Point D (at 1)
Why: More likely means probability > 0.5 and < 1.

Hard (10)

A bag contains 6 red, 4 blue, and 2 green marbles. If one marble is drawn at random, what is the probability that it is NOT blue?
Why: Probability of not blue = (Red + Green) / Total = (6+2)/12 = 8/12 = 2/3.
Two coins are tossed simultaneously. What is the probability of getting exactly one head?
Why: Sample space is {HH, HT, TH, TT}. Exactly one head is {HT, TH}. So 2 out of 4 outcomes.
A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a face card (King, Queen, or Jack)?
Why: There are 12 face cards (3 per suit) in a deck of 52 cards. So 12/52.
In a survey of 200 students, 120 students like Maths, 80 like Science, and 40 like both. If a student is chosen randomly, what is the probability that the student likes Maths but not Science?
Why: Students liking Maths only = (Students liking Maths) - (Students liking both) = 120 - 40 = 80. Probability = 80/200 = 0.4.
Assertion (A): The probability of an event cannot be negative. Reason (R): The number of favorable outcomes and the total number of outcomes are always non-negative.
Why: Probability is a ratio of counts, which are non-negative, hence probability cannot be negative.
Assertion (A): If a coin is tossed 1000 times and heads appears 500 times, the experimental probability of heads is 0.5. Reason (R): Theoretical probability always equals experimental probability in the long run.
Why: Experimental probability is calculated correctly, but theoretical and experimental probabilities only tend to converge in the long run, not always equal.
A survey was conducted among 150 students about their favorite fruit. 60 students liked apples, 45 liked bananas, and 30 liked oranges. The remaining students liked grapes. If a student is chosen at random, what is the probability that they like grapes?
- 0.1 (correct)
- 0.2
- 0.25
- 0.3
Why: Students liking grapes = 150 - (60+45+30) = 15. Probability = 15/150 = 0.1.
A box contains 10 tickets, numbered 1 to 10. If two tickets are drawn at random without replacement, what is the probability that both tickets have odd numbers?
- 5/18
- 2/9 (correct)
- 1/4
- 1/5
Why: P(1st odd) = 5/10. P(2nd odd | 1st odd) = 4/9. Multiply them.
A tree diagram shows the outcomes of tossing a coin twice. What is the probability of getting at least one head?
- 1/4
- 1/2
- 3/4 (correct)
- 1
Why: Outcomes with at least one head are HH, HT, TH. That's 3 out of 4 total outcomes.
A bag contains 3 red, 2 blue, and 5 green marbles. A marble is drawn, its color noted, and then replaced. A second marble is drawn. Using a tree diagram, find the probability that both marbles drawn are blue.
- 1/25 (correct)
- 4/100
- 1/50
- 2/10
Why: P(1st blue) = 2/10. P(2nd blue) = 2/10 (with replacement). Multiply them.