A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel is (A) (

NCERT Exemplar · Q26

\frac{1}{3}

Sikkim CbseMCQ· 1mImportance★★★★★est

82% · 76/93 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Classical Probability

Classical Probability: The "Fair Game" Definition

Imagine you roll a fair six-sided die. Before it lands, you know there are exactly six possible outcomes — 1, 2, 3, 4, 5, or 6 — and you have no reason to believe any one face is more likely than another. That gut feeling of "all outcomes are equally likely" is the entire foundation of classical probability.

The Intuition

Classical probability was born from games of chance — dice, coins, cards. In these settings, the physical symmetry of the objects (a balanced die, a fair coin) guarantees that no outcome is favoured. So the probability of an event is simply:

Number of ways the event can happen, divided by the total number of possible outcomes.

If you want the chance of rolling an even number on a die, count the evens: 2, 4, 6 — that's 3 ways. Total outcomes: 6. So probability = $3/6 = 1/2$ .

This is the "counting" approach. It works beautifully when the underlying experiment is symmetric and finite.

The Precise Statement

P(E) = \frac{\text{Number of outcomes favourable to event } E}{\text{Total number of equally likely outcomes}}

This is called the classical definition (or a priori definition) of probability. It was formalised by Pierre-Simon Laplace in the 18th century.

Three conditions must hold for this definition to apply:

Finite sample space — there are only a fixed, countable number of possible outcomes.
Equally likely outcomes — each outcome has the same chance of occurring (the "fairness" condition).
Mutually exclusive outcomes — no two outcomes can happen at the same time.

Watch out

The biggest mistake students make is applying classical probability to situations where outcomes are not equally likely. For example: "I can either pass or fail the exam — two outcomes, so probability of passing is $1/2$ ." That's nonsense, because passing and failing are not equally likely. The die works only because the die is fair.

A Simple Example

Problem: A bag contains 3 red marbles and 2 blue marbles. You pick one marble at random. What is the probability it is red?

Step 1 — Identify the sample space: There are 5 marbles total. If the marbles are physically identical except for colour, and you pick without looking, each marble is equally likely to be chosen. So total outcomes = 5.

Step 2 — Count favourable outcomes: 3 marbles are red. So favourable outcomes = 3.

Step 3 — Apply the formula:

P(\text{red}) = \frac{3}{5}

That's it. No deeper theory needed for this case.

When Classical Probability Fails

Classical probability cannot handle: …

Start your 14-day free trial to unlock the full solution →

To find the probability of selecting a vowel from 'PROBABILITY', we count the total letters (11) and the vowels (4), then apply the classical probability formula to get $\boxed{\frac{4}{11}}$ .

When we talk about probability in simple scenarios like this, we often use what's called Classical Probability. This approach is applicable when all possible outcomes of an experiment are equally likely to occur. For instance, if you roll a fair die, each face (1, 2, 3, 4, 5, 6) has an equal chance of appearing. Similarly, when you select a letter at random from a word, each letter position has an equal chance of being chosen.

The core idea is to determine two things:

The total number of distinct, equally likely outcomes possible. This forms our sample space.
The number of outcomes that satisfy a specific condition (our "event"). These are the favorable outcomes.

The probability of the event is then simply the ratio of favorable outcomes to the total number of outcomes.

E

Let's apply this to the given problem.

Identify the total number of possible outcomes.

The word is 'PROBABILITY'. We need to count the total number of letters in this word.

P, R, O, B, A, B, I, L, I, T, Y

Counting them, we find there are 11 letters. Each of these 11 letters has an equal chance of being selected.

So, the total number of possible outcomes is 11.
Identify the number of favorable outcomes.

We are interested in the probability that the selected letter is a vowel. The vowels in the English alphabet are A, E, I, O, U.

Let's list the letters in 'PROBABILITY' and identify the vowels among them:
- P (consonant)
- R (consonant)
- O (vowel)
- B (consonant)
- A (vowel)
- B (consonant)
- I (vowel) …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ25 — 6 boys and 6 girls sit in a row at random. The probability that all th…→Next questionQ27 — If the probabilities for A to fail in an examination is 0.2 and that f…

Q.A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel is (A) 13\frac{1}{3}31​ (B) 411\frac{4}{11}114​ (C) 211\frac{2}{11}112​ (D) 311\frac{3}{11}113​

🎓Unlock everything free for 14 days

Q.A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel is
(A) $\frac{1}{3}$
(B) $\frac{4}{11}$
(C) $\frac{2}{11}$
(D) $\frac{3}{11}$

Unlock everything free for 14 days