There are four men and six women on the city council. If one council member is selected for a committee at ran

Exercise 14.2 · Q6

Q. There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?

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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 selecting one person at random from a group of 4 men and 6 women, the probability of choosing a woman is the ratio of women to total members: $\frac{6}{10} = \frac{3}{5}$ or $60\%$ .

Why this approach works

Classical probability rests on a simple principle: when all outcomes are equally likely, the probability of an event is the fraction of favorable outcomes among all possible outcomes. Here, each council member has an equal chance of being selected, so we count how many ways we can pick a woman and divide by the total number of people we could pick.

The key insight is recognizing that "selecting one person at random" means every individual has the same $\frac{1}{10}$ chance of being chosen. We then group these individual outcomes by the characteristic we care about—gender in this case.

Step-by-step solution

Count the total number of council members.

We have 4 men and 6 women, giving us $4 + 6 = 10$ people total.
Identify the favorable outcomes.

The question asks for the probability that the selected member is a woman. There are 6 women on the council, so there are 6 favorable outcomes.
Apply the classical probability formula.

$P(\text{woman selected}) = \frac{\text{number of women}}{\text{total number of members}} = \frac{6}{10}$ …

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Q.There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?

Why this approach works

Step-by-step solution

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