NCERT Solutions for Class 9 Maths Chapter 13: Probability — Free PDF

Chapter 13 introduces the concept of probability through an experimental (empirical) approach. Unlike the theoretical probability in Class 10 (which uses equally likely outcomes), this chapter focuses on probability based on actual experiments and observations.

Key topics:
- Random experiments and outcomes
- Empirical (experimental) probability
- Frequency-based probability calculations
- The range of probability: $0 \leq P(E) \leq 1$
- Sum of probabilities of all outcomes equals 1

The chapter has one exercise with practical, data-based problems.

Probability is one of the most widely used branches of mathematics. Weather forecasts, medical diagnoses, insurance premiums, game strategies, and quality control in factories all rely on probability. In Class 9, the NCERT textbook focuses on the empirical approach because it connects probability to real data that students can collect and analyse. This builds strong intuition before the more abstract theoretical approach in Class 10.

Experiment: Any process that produces a well-defined result. For example, tossing a coin, rolling a die, or recording the weather.

Random experiment: An experiment whose outcome cannot be predicted with certainty in advance. Each repetition may produce a different result.

Trial: A single performance of a random experiment. If you toss a coin $100$ times, each toss is one trial.

Outcome: A possible result of a trial. For a coin toss, the outcomes are Head and Tail.

Event: A collection of one or more outcomes. "Getting an even number" when rolling a die is an event consisting of the outcomes $\{2, 4, 6\}$.

Empirical (experimental) probability: Probability calculated from observed data.

Theoretical probability (covered in Class 10): Probability calculated by assuming equally likely outcomes.

Key properties:
- $0 \leq P(E) \leq 1$ for any event $E$
- $P(\text{certain event}) = 1$ (event that always happens)
- $P(\text{impossible event}) = 0$ (event that never happens)
- $P(E) + P(\bar{E}) = 1$, where $\bar{E}$ means "event $E$ does not occur"
- The sum of empirical probabilities of all outcomes equals $1$

Law of large numbers: As the number of trials increases, the empirical probability gets closer to the theoretical probability. This is why we need many trials for accurate estimates.

Problem 1: A coin is tossed 1000 times with the following frequencies: Head = 455, Tail = 545. Compute the probability of each event.

Solution:

Verification: $0.455 + 0.545 = 1$ ✓

Note that the empirical probability of Head is close to but not exactly $0.5$. With more trials, it would get even closer to $0.5$.

Problem 2: Two coins are tossed simultaneously 500 times and the outcomes are:

Find the probability of each outcome.

Solution:

Verification: $0.21 + 0.55 + 0.24 = 1$ ✓

Theoretical probabilities would be $0.25$, $0.50$, and $0.25$ respectively. The empirical values are reasonably close.

Problem 3: A die is thrown 1000 times with the following outcomes:

Find the probability of getting: (i) 3 (ii) a number less than 4 (iii) a number greater than or equal to 4.

Solution:
(i) $P(3) = \frac{157}{1000} = 0.157$

(ii) Numbers less than 4: $\{1, 2, 3\}$

(iii) Numbers $\geq 4$: $\{4, 5, 6\}$

Alternatively: $P(\geq 4) = 1 - P(< 4) = 1 - 0.486 = 0.514$ ✓

Using the complement is often the faster method when the complementary event involves fewer outcomes.

Problem 4: In a cricket match, a batsman hits a boundary 6 times out of 30 balls. Find the probability that he did NOT hit a boundary.

Solution:

Problem 5: 1500 families were surveyed for the number of children:

(i) Find $P(\text{2 children})$. (ii) Find $P(\text{at most 2 children})$. (iii) Find $P(\text{more than 2 children})$.

Solution:
(i) $P(\text{2 children}) = \frac{365}{1500} = \frac{73}{300} \approx 0.243$

(ii) At most 2 children means 0, 1, or 2:

(iii) More than 2 children means 3 or 4+:

Alternatively: $P(> 2) = \frac{82 + 28}{1500} = \frac{110}{1500} = \frac{11}{150}$ ✓

Problem 6: A bag has 3 red, 5 blue, and 2 green balls. A ball is drawn and the colour noted, then put back. After 1000 draws: Red = 300, Blue = 500, Green = 200. Calculate the probability of each.

Solution:

Note: These empirical probabilities are close to the theoretical values $\frac{3}{10}, \frac{5}{10}, \frac{2}{10}$, which is expected with a large number of trials.

Example 1: Defective items in a factory

A factory produces $500$ bulbs in a day. Quality testing reveals that $25$ are defective. If a bulb is selected at random from the day's production, find the probability that it is (a) defective, (b) not defective.

Solution:
(a) $P(\text{defective}) = \frac{25}{500} = \frac{1}{20} = 0.05$

(b) $P(\text{not defective}) = 1 - 0.05 = 0.95$

This means $95\%$ of the bulbs are expected to work properly.

Example 2: Weather data

Over the past $365$ days, it rained on $73$ days. What is the empirical probability that it will rain on a randomly chosen day?

So based on past data, there is a $20\%$ chance of rain on any given day.

Example 3: Blood group probability

In a sample of $200$ people, blood groups are distributed as: A = 40, B = 56, AB = 16, O = 88. Find the probability that a randomly selected person has blood group B or AB.

Example 4: Using complements efficiently

A student answers $90$ out of $120$ questions correctly. Find the probability of answering a question incorrectly.

Mistake 1: Confusing empirical and theoretical probability.
In Class 9, you compute probability from experimental data (frequencies). Do not use the theoretical formula $\frac{\text{favourable outcomes}}{\text{total outcomes}}$ unless the problem specifically states equally likely outcomes. For example, if a coin is tossed $100$ times and heads appears $48$ times, write $P(H) = \frac{48}{100} = 0.48$, not $P(H) = \frac{1}{2}$.

Mistake 2: Writing probability greater than 1 or less than 0.
Probability is always between $0$ and $1$. If your calculation gives $P(E) > 1$ or $P(E) < 0$, you have made an error. Go back and check the numerator and denominator.

Mistake 3: Forgetting to verify that probabilities sum to 1.
After computing the probability of every possible outcome, add them up. The total must be $1$. If it is not, you have either missed an outcome or made an arithmetic error.

Mistake 4: Using the wrong denominator.
The denominator is always the total number of trials, not the number of favourable outcomes or the number of categories. For example, if $1500$ families are surveyed, every probability calculation uses $1500$ in the denominator.

Mistake 5: Not simplifying fractions.
Always simplify your answer. Write $\frac{1}{5}$ rather than $\frac{6}{30}$. In board exams, unsimplified fractions may lose presentation marks.

Q1. A coin is tossed $200$ times. Heads appear $112$ times. Find $P(\text{Head})$ and $P(\text{Tail})$.

Q2. In a class of $50$ students, $15$ like cricket, $20$ like football, and $15$ like basketball. A student is chosen at random. Find the probability that the student likes football.

Q3. A die is thrown $600$ times. The outcomes are: 1 (98), 2 (102), 3 (95), 4 (105), 5 (100), 6 (100). Find the probability of getting an even number.

Q4. Out of $400$ light switches tested, $16$ were found faulty. Find the probability that a switch chosen at random is not faulty.

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Answers:

A1. $P(H) = \frac{112}{200} = \frac{14}{25} = 0.56$. $P(T) = 1 - 0.56 = 0.44$.

A2. $P(\text{football}) = \frac{20}{50} = \frac{2}{5} = 0.4$.

A3. Even numbers: $\{2, 4, 6\}$. Frequency $= 102 + 105 + 100 = 307$. $P(\text{even}) = \frac{307}{600} \approx 0.512$.

A4. $P(\text{not faulty}) = 1 - \frac{16}{400} = 1 - \frac{1}{25} = \frac{24}{25} = 0.96$.