If the probabilities for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that e

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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: …

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When two events can overlap, the probability that at least one occurs is found using the inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . Since we don't know whether A and B's failures are independent, the answer depends on their relationship, making the probability at most $0.5$ .

The question asks for the probability that "either A or B fails," which in probability language means "A fails OR B fails" — the union of two events.

The natural instinct might be to simply add the individual probabilities: $0.2 + 0.3 = 0.5$ . But this overlooks a crucial subtlety: what if both A and B fail together? If we just add, we've counted that scenario twice.

The correct formula for the probability of a union is:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

This subtracts the overlap to avoid double-counting.

Let me work through what we know and what we can deduce:

Given information: $P(\text{A fails}) = 0.2$ and $P(\text{B fails}) = 0.3$ .
What we need: $P(\text{A fails OR B fails})$ .
Apply the union formula:

P(\text{A fails OR B fails}) = 0.2 + 0.3 - P(\text{both fail})

= 0.5 - P(\text{both fail})

The key constraint: The probability that both fail, $P(A \cap B)$ , must be non-negative (probabilities can't be negative). So:

P(A \cap B) \geq 0

Therefore: $P(\text{A fails OR B fails}) = 0.5 - P(A \cap B) \leq 0.5$ …

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Q.If the probabilities for A to fail in an examination is 0.20.20.2 and that for B is 0.30.30.3, then the probability that either A or B fails is (A) >.5> .5>.5 (B) .5.5.5 (C) ≤.5\leq .5≤.5 (D) 000

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Q.If the probabilities for A to fail in an examination is $0.2$ and that for B is $0.3$ , then the probability that either A or B fails is
(A) $> .5$
(B) $.5$
(C) $\leq .5$
(D) $0$

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