Q.In a group of 100 sports car buyers, 40 bought alarm systems, 30 purchased bucket seats, and 20 purchased an alarm system and bucket seats. If a car buyer chosen at random, bought an alarm system, what is the probability they also bought bucket seats?

Event and Its Probability

Probability is the language of uncertainty. When you toss a coin, you don't know if it will land heads or tails — but you do know that each outcome is equally likely. That feeling of "how likely" is what probability measures.

The Intuition: What Does "Probability" Mean?

Imagine you roll a fair six-sided die. Before it stops, you can't say which number will show up. But you can say this: if you roll it many, many times, each face will appear roughly one-sixth of the time. That fraction — $\frac{1}{6}$ — is the probability of getting, say, a 4.

So probability is a number between 0 and 1 that tells you how often something happens in the long run. A probability of 0 means it never happens; a probability of 1 means it always happens.

The Building Blocks: Sample Space and Event

Before we can talk about probability, we need two things:

1. Sample space ($S$) — the set of all possible outcomes of an experiment. For a die roll, $S = \{1,2,3,4,5,6\}$.

2. Event ($E$) — any subset of the sample space. An event is a collection of outcomes you care about. For example, "rolling an even number" is the event $E = \{2,4,6\}$.

The Precise Definition

For a fair experiment — where every outcome in the sample space is equally likely — the probability of an event $E$ is:

This is the classical definition of probability. It works perfectly for dice, coins, cards, and any situation where symmetry makes all outcomes equally likely.

For the die example:

• $S$ has 6 outcomes.
• $E = \{2,4,6\}$ has 3 outcomes.
• So $P(\text{even number}) = \frac{3}{6} = \frac{1}{2}$.

Why This Makes Sense

The fraction $\frac{3}{6}$ isn't just a formula — it's a direct translation of the intuition. If you roll a fair die 600 times, you expect about 300 evens. The probability $\frac{1}{2}$ is the theoretical prediction of that long-run relative frequency.

A Quick Example

Problem: A bag contains 3 red marbles and 5 blue marbles. You pick one marble at random. What is the probability it is red?

Step 1: Sample space $S$ has $3 + 5 = 8$ marbles — 8 equally likely outcomes.

Step 2: Event $E$ = "red marble" has 3 outcomes.

Step 3: $P(E) = \frac{3}{8}$.

That's it. The probability is $\frac{3}{8}$ — a number between 0 and 1 that tells you how likely a red marble is.

The Big Picture

Probability is just counting, carefully. You count how many ways your event can happen, divide by how many things can happen in total, and you get a number that captures the chance. Every more advanced idea — conditional probability, Bayes' theorem, random variables — builds on this simple foundation.

The final answer: For a fair experiment with equally likely outcomes, the probability of an event $E$ is $\frac{\text{number of outcomes in } E}{\text{total number of outcomes in the sample space}}$.