14.1. Probability — A Theoretical Approach

The theory of probabilities and the theory of errors now constitute a formidable body of great mathematical interest and of great practical importance.

This introductory statement underlines the significance of probability theory both mathematically and practically. It highlights how probability is not just an abstract concept but has real-world applications, such as in risk assessment, statistics, and various fields of science.

Think of probability as a tool used by weather forecasters. They use probability to predict the likelihood of rain based on atmospheric data, helping people decide whether to carry an umbrella.

Suppose a coin is tossed at random. When we speak of a coin, we assume it to be ‘fair’, that is, it is symmetrical so that there is no reason for it to come down more often on one side than the other. We call this property of the coin as being ‘unbiased’. By the phrase ‘random toss’, we mean that the coin is allowed to fall freely without any bias or interference.

A fair coin means that the chances of it landing on heads or tails are equal (50% for each side). This concept is crucial in probability because it allows predictions about outcomes based on known equal chances. A random toss allows for no external factors that could influence the result, ensuring each outcome is purely based on chance.

Imagine tossing a coin in a game. If players believe it's fair, they will trust the outcome. If it were rigged to favor heads, they might not trust the game, highlighting the importance of fairness in probability.

We can reasonably assume that each outcome, head or tail, is as likely to occur as the other. We refer to this by saying that the outcomes head and tail are equally likely.

The term 'equally likely outcomes' refers to scenarios where each outcome has the same probability of occurring. This principle simplifies calculations in probability, allowing us to apply a common formula for events with an equal chance of happening.

Think of rolling a fair six-sided die. Each side (1 through 6) has the same chance of landing face up. You can imagine it as having six identical doors; each time you choose a door randomly, you would expect each one to open just as often as the others, demonstrating equally likely outcomes.

However, in this chapter, from now on, we will assume that all the experiments have equally likely outcomes.

Unlike the coin toss, not all scenarios have equally likely outcomes. For instance, drawing a ball from a bag with 4 red balls and 1 blue ball means that the probability of picking a red ball is higher than that of picking a blue ball. This section emphasizes the need to analyze the experiment to understand outcomes before applying probability rules.

If you have a bag of apples and one spoiled apple, when you reach in without looking, it’s more likely you’ll grab a good apple simply because there are more of them. This situation shows how outcomes can be unevenly distributed.

The theoretical probability (also called classical probability) of an event E, written as P(E), is defined as Number of outcomes favourable to E / Number of all possible outcomes of the experiment where we assume that the outcomes of the experiment are equally likely.

The theoretical probability formula allows us to calculate the likelihood of an event based on known outcomes. Here, P(E) is the probability of event E occurring. The key point is that this only applies when outcomes are equally likely. This mathematical approach enables clear predictions about future probabilities from established ratios.

Consider a box with 10 apples – 7 are red and 3 are green. If you randomly pick an apple, the theoretical probability of picking a red one is 7 out of 10, or 70%. This simply illustrates how you can use ratios to predict outcomes in everyday scenarios.

This definition of probability was given by Pierre Simon Laplace in 1795. Probability theory had its origin in the 16th century when an Italian physician and mathematician J.Cardan wrote the first book on the subject, The Book on Games of Chance.

Understanding the history of probability showcases its evolution as a field of study. It started from practical concerns about gambling and has developed into a comprehensive theory applied across various disciplines. Knowing how great minds contributed to probability enriches our appreciation of its mathematical foundations.

It’s like how cooking recipes have evolved from simple techniques to complex culinary practices. Just as chefs have learned from each other, mathematical thought in probability has built upon the ideas of previous scholars.

In recent years, probability has been used extensively in many areas such as biology, economics, genetics, physics, sociology etc.

Probability is foundational in numerous fields for decision-making and predictions. In biology, it helps analyze genetic variations; in economics, it assesses risks and investments; while in physics, it explains uncertainty in particle behavior. Its wide applicability underscores the relevance of mastering this concept.

Think of a weather app using probability to predict rain chances. If it tells you there’s a 70% chance of rain, understanding probability allows you to make informed choices about carrying an umbrella.

Example 1: Find the probability of getting a head when a coin is tossed once. Also find the probability of getting a tail. Solution: In the experiment of tossing a coin once, the number of possible outcomes is two — Head (H) and Tail (T). Let E be the event ‘getting a head’. The number of outcomes favourable to E, (i.e., of getting a head) is 1. Therefore, P(E) = 1/2. Similarly, if F is the event ‘getting a tail’, then P(F) = 1/2.

This example illustrates how to apply the theoretical probability formula to a simple experiment - tossing a coin. There are two equally likely outcomes, and calculating the probability involves identifying favorable outcomes versus total outcomes. Each outcome has a probability of 1/2, demonstrating straightforward calculation of probability.

When flipping a coin for a simple choice, like deciding who goes first in a game, you trust that it's 50/50, which is reflected in the calculated probabilities. It's a common decision-making tool applied in various simple scenarios.

Remarks: An event having only one outcome of the experiment is called an elementary event. In Example 1, both the events E and F are elementary events. Similarly, in Example 2, all the three events, Y, B and R are elementary events. In Example 1, we note that: P(E) + P(F) = 1.

Elementary events occur when there is only one possible outcome for an event. Understanding elementary events is crucial because they simplify the concept of probability, as their probabilities will sum up to 1 when considering all possible outcomes from a single experiment. This fundamental concept is essential for building upon more complex probability problems later.

If you think of flipping a coin, both heads and tails are exclusive events that can't occur simultaneously. If you know one occurred, the other did not, illustrating how probabilities sum up to total certainty.

In general, it is true that for an event E, P(E) + P(E') = 1, where E' stands for ‘not E’. We also say that E and E' are complementary events.

Complementary events refer to two mutually exclusive outcomes where one event occurring means the other cannot occur. The relation P(E) + P(E') = 1 emphasizes that knowing the probability of one event automatically defines the other. This relationship deepens the understanding of probability as a whole.

Imagine a light switch with two positions: on and off. If you know the switch is on, you know for sure it’s not off. Similarly, in probability, if you know the chance of rain today is 30%, you can deduce that there’s a 70% chance it won't rain, representing complementary events.

--