Probability — Study Notes

1. Introduction

Every day we use the language of chance without realising it. When we say "it will probably rain today", "the chance of India winning is high", or "it is unlikely that the bus will be late", we are talking about probability. Probability is the branch of mathematics that measures how likely an event is to occur. It gives a numerical value to uncertainty, turning vague statements of chance into precise numbers between 0 and 1.

The modern study of probability began with problems related to games of chance, such as dice and cards, in the seventeenth century. Mathematicians like Blaise Pascal and Pierre de Fermat developed the basic rules by analysing gambling games. In this chapter we will study the empirical (experimental) approach to probability, which is based on actually performing experiments and observing outcomes. We will learn how to compute the probability of an event as the ratio of favourable outcomes to the total number of trials.

2. Experiments and Events

Random Experiment

An experiment is called a random experiment if its outcome cannot be predicted in advance. For example, tossing a coin, throwing a die, drawing a card from a pack, or counting the number of vehicles passing a point in an hour are all random experiments. Although we cannot predict a single outcome, we can list all the possible outcomes.

Trial and Outcome

A trial is a single performance of a random experiment, and the result of a trial is called an outcome. For example, tossing a coin once is a trial, and getting "head" is an outcome.

Events

An event is a collection of outcomes. For example, when a die is thrown, "getting an even number" is an event (with outcomes 2, 4 and 6), and "getting a number greater than 4" is another event (with outcomes 5 and 6).

3. Empirical Probability

The Probability Formula

Suppose an experiment is repeated n times. Let the event E occur m times in these n trials. Then the empirical (experimental) probability of the event E is:

P(E) = (number of trials in which E occurs) / (total number of trials) = m/n

For example, if a coin is tossed 100 times and heads appear 52 times, then the probability of getting a head, based on this experiment, is 52/100 = 0.52.

Properties of Probability

• The probability of an event always lies between 0 and 1, inclusive: 0 <= P(E) <= 1.
• If an event is impossible, its probability is 0.
• If an event is certain, its probability is 1.
• The sum of the probabilities of all the outcomes of an experiment is 1.
• Since m cannot exceed n, the ratio m/n can never be greater than 1.

Probability of the Complement

If E is an event, the event "not E" is called the complement of E. The sum of the probabilities of an event and its complement is 1: P(E) + P(not E) = 1.

For example, if P(getting a head) = 0.52, then P(getting a tail) = 1 - 0.52 = 0.48.

4. Important Points about Empirical Probability

Empirical Versus Theoretical Probability

The probability we compute from actual experiments is called empirical or experimental probability. It is based on observation. When an experiment is repeated a very large number of times, the empirical probability of an event approaches its theoretical probability, but the two may differ for a small number of trials.

The Role of Large Numbers

The more trials we perform, the more reliable the empirical probability becomes. This is sometimes called the law of large numbers. For example, the empirical probability of getting a head from 10 tosses may differ noticeably from 0.5, but from 1000 tosses it will be very close to 0.5.

Equal Chance vs Frequency

Empirical probability does not assume that all outcomes are equally likely; it simply records what actually happened in the trials. This makes it different from the classical probability we will study in later classes, where we assume equally likely outcomes.

5. Solved Examples and Applications

Example 1: Coin Toss

A coin is tossed 500 times and heads occur 250 times. The probability of getting a head is 250/500 = 0.5, and the probability of getting a tail is also 0.5.

Example 2: Marks of Students

In a survey, 35 out of 50 students of a class passed an examination. The empirical probability that a student of that class passed is 35/50 = 0.7, and the probability of failing is 15/50 = 0.3. Note that 0.7 + 0.3 = 1.

Applications

Probability is used in weather forecasting, insurance, games, quality control in factories, medical testing and many other fields. In each case, past data (relative frequency) is used to estimate the chance of future events.

Quick Revision Tables

Table 1: Basic Terms of Probability

Table 2: Probability Values and Properties

Mind Map

Important Diagrams (SVG)

Diagram 1: The Probability Scale

Diagram 2: Coin Toss Experiment

Common Mistakes

1. Students often write probabilities greater than 1 or less than 0. Since P(E) = m/n with m at most n, the value must always lie between 0 and 1.
2. A common error is dividing the total number of trials by the number of favourable trials, inverting the formula. Always compute favourable/total.
3. Students forget that the probability of a certain event is 1 and of an impossible event is 0, and instead guess intermediate values.
4. Students confuse empirical probability (based on experiments) with theoretical probability (based on equally likely outcomes); they must state which one they are computing.
5. A frequent mistake is using favourable outcomes without counting the total number of trials correctly from the given data.
6. Students forget the complement rule: P(not E) = 1 - P(E), and instead add probabilities wrongly.
7. Students assume that the empirical probability of a head is exactly 1/2 for a small number of tosses; it may differ and approaches 1/2 only for many trials.
8. Students write the probability as a fraction without simplifying, or as m/n without mentioning what m and n stand for.

Exam Tips

1. Always define m (number of favourable outcomes) and n (total number of trials) before applying the formula P(E) = m/n.
2. State clearly that the probability lies between 0 and 1 and use the complement rule when asked for the probability of "not an event".
3. Read the question carefully to determine whether it asks for empirical or theoretical probability and answer accordingly.
4. Simplify fractions in the final answer, and give the value as a decimal if asked.
5. When the total number of trials and the frequency of one event are given, find the frequency of the complementary event by subtraction.
6. Practise reading data from tables and counting favourable outcomes, as most exam questions present data in tables.
7. Mention the property "sum of probabilities of all outcomes is 1" when verifying answers.

Conclusion

In this chapter we were introduced to probability, the mathematics of chance. We learnt the basic vocabulary of random experiments, trials, outcomes and events, and we defined the empirical probability of an event as the ratio of the number of favourable outcomes to the total number of trials. We studied the fundamental properties of probability, including the fact that it always lies between 0 and 1, that a certain event has probability 1, and that the probability of an event plus the probability of its complement equals 1. We also understood the law of large numbers, which says that empirical probability becomes more accurate with more trials. These ideas lay the foundation for the theoretical probability that we will study in higher classes, and they have applications in science, business, games and everyday decision making.