Match the proposed probability under Column $C_1$ with the appropriate written description under column $C_2$: | $C_1$ (Probability) | $C_2$ (Written Description) | | --- | --- | | (a) $0.95$ | (i) An incorrect assignment | | (b) $0.02$ | | (ii) No chance of happening (c) $-0.3$ (iii) As much chance of happening as not. (d) $0.5$ (iv) Very likely to happen (e) $0$ (v) Very little chance of happening

> [!TLDR] > Probabilities must lie in $[0, 1]$; values near $0$ mean unlikely, near $1$ mean likely, and $0.5$ means equally likely outcomes. Matching: (a)–(iv), (b)–(v), (c)–(i), (d)–(iii), (e)–(ii). The foundation here is the **axiomatic definition of probability**: for any event $A$, we require $0 \le P(A) \le 1$. This isn't arbitrary—it reflects the fact that probability measures certainty on a scale from impossible ($0$) to certain ($1$). Any value outside this interval violates the axioms and cannot represent a valid probability. With that framework, we interpret each numerical value by its position on the $[0, 1]$ scale and match it to the qualitative description. ## Step-by-step matching 1. **(a) $P = 0.95$** A probability of $0.95$ means the event happens in $95$ out of $100$ trials on average. This is very close to certainty—only a $5\%$ chance of *not* happening. The natural description is **(iv) Very likely to happen**. 2. **(b) $P = 0.02$** Here $P = 0.02 = 2\%$, so the event occurs roughly $2$ times in $100$ trials. It's possible but rare. This matches **(v) Very little chance of happening**. 3. **(c) $P = -0.3$** A negative probability has no meaning in standard probability theory—it violates the non-negativity axiom. This is **(i) An incorrect assignment**. > [!WARNING] > Negative probabilities sometimes appear in quantum mechanics (quasi-probability distributions) or as intermediate steps in calculations, but in elementary probability they signal an error. 4. **(d) $P = 0.5$** …

Match the proposed probability under Column C₁ wit – Class 11 Mathematics

NCERT Exemplar · Q42

Match the proposed probability under Column $C_1$ with the appropriate written description under column $C_2$ :

$C_1$ (Probability)	$C_2$ (Written Description)
(a) $0.95$	(i) An incorrect assignment
(b) $0.02$

(ii) No chance of happening (c) $-0.3$ (iii) As much chance of happening as not. (d) $0.5$ (iv) Very likely to happen (e) $0$ (v) Very little chance of happening

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Concept understanding — Probability Axioms

Probability Axioms: From Intuition to Precision

Imagine you're rolling a fair six-sided die. Before you throw it, you know a few things for certain: the result will be one of the numbers 1 through 6. You also know that some outcomes are equally likely — each face has a 1-in-6 chance. And you know that the chance of getting either a 1 or a 2 is simply the sum of their individual chances: $\frac{1}{6} + \frac{1}{6} = \frac{1}{3}$ .

These three ideas — that probabilities are numbers between 0 and 1, that something must happen (total probability = 1), and that probabilities of non-overlapping events add — are the bedrock of all probability theory. They are so fundamental that we call them axioms: self-evident truths from which everything else is derived.

The Three Axioms (Kolmogorov's Axioms)

Let’s make this precise. We have a sample space $S$ — the set of all possible outcomes. An event $A$ is any subset of $S$ (like "rolling an even number" = $\{2,4,6\}$ ). The probability of an event $A$ is written $P(A)$ .

P(A) \geq 0 \quad \text{for every event } A

Axiom 1 (Non-negativity): A probability can never be negative. This matches our intuition: you can't have a "less than zero" chance of something happening. The smallest possible probability is 0 (an impossible event).

P(S) = 1

Axiom 2 (Normalization): The probability that some outcome in the sample space occurs is exactly 1. Something must happen. This is why we say "the die will show 1,2,3,4,5, or 6" with certainty.

A

Axiom 3 (Additivity): For events that don't overlap, the probability of "A or B" is just the sum of their individual probabilities. This is why the chance of rolling a 1 or a 2 is $\frac{1}{6} + \frac{1}{6}$ .

Watch out

This additivity only works for mutually exclusive events. If events can happen together (like "rolling an even number" and "rolling a number greater than 3"), you cannot simply add their probabilities — you'd double-count the overlap.

Why These Three Are Enough

From these three simple rules, we can derive everything else in probability. For example:

Complement rule: $P(\text{not } A) = 1 - P(A)$ . Why? Because $A$ and "not $A$ " are mutually exclusive and together cover the whole sample space. By Axiom 3: $P(A) + P(\text{not } A) = P(S) = 1$ .
Probability of an impossible event: $P(\emptyset) = 0$ . Since $S$ and $\emptyset$ are mutually exclusive and $S \cup \emptyset = S$ , we get $P(S) + P(\emptyset) = P(S)$ , so $P(\emptyset) = 0$ . …

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Probabilities must lie in $[0, 1]$ ; values near $0$ mean unlikely, near $1$ mean likely, and $0.5$ means equally likely outcomes. Matching: (a)–(iv), (b)–(v), (c)–(i), (d)–(iii), (e)–(ii).

The foundation here is the axiomatic definition of probability: for any event $A$ , we require $0 \le P(A) \le 1$ . This isn't arbitrary—it reflects the fact that probability measures certainty on a scale from impossible ( $0$ ) to certain ( $1$ ). Any value outside this interval violates the axioms and cannot represent a valid probability.

With that framework, we interpret each numerical value by its position on the $[0, 1]$ scale and match it to the qualitative description.

Step-by-step matching

(a) $P = 0.95$

A probability of $0.95$ means the event happens in $95$ out of $100$ trials on average. This is very close to certainty—only a $5\%$ chance of not happening. The natural description is (iv) Very likely to happen.
(b) $P = 0.02$

Here $P = 0.02 = 2\%$ , so the event occurs roughly $2$ times in $100$ trials. It's possible but rare. This matches (v) Very little chance of happening.
(c) $P = -0.3$

A negative probability has no meaning in standard probability theory—it violates the non-negativity axiom. This is (i) An incorrect assignment.

Watch out

Negative probabilities sometimes appear in quantum mechanics (quasi-probability distributions) or as intermediate steps in calculations, but in elementary probability they signal an error.

(d) $P = 0.5$ …

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