14. PROBABILITY

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

This introduces us to the concept of probability, describing it as a significant mathematical theory that plays a critical role in various practical applications. It indicates that understanding probability is essential not just in mathematics, but across multiple fields where uncertainty and chance are involved.

Imagine you're analyzing weather patterns; understanding probability helps you predict the chances of rain, making decisions about whether to carry an umbrella or not.

Let us consider the following situation: Suppose a coin is tossed at random... We refer to this by saying that the outcomes head and tail, are equally likely.

When tossing a fair coin, there are two possible outcomes: heads or tails. The term 'equally likely' means that both outcomes have the same probability of occurring. This basic example serves as a foundation for understanding more complex probability situations.

Think of flipping a coin as a simple game you might play. Since there's no bias, you're just as likely to get heads as you are tails each time you flip it.

Are the outcomes of every experiment equally likely? ... we will assume that all the experiments have equally likely outcomes.

Not all experiments have equally likely outcomes. For example, if you draw a ball from a bag containing different colored balls, the probability of drawing a specific color depends on how many of each color are present. However, in this analysis, we will simplify our models by assuming that outcomes are equally likely.

Imagine a jar filled with different colored marbles. If you know there are more red than blue, you're more likely to pull a red one out than a blue one. This illustrates that not all outcomes in experiments yield equal chances.

In Class IX, we defined the experimental or empirical probability P(E) of an event E as Number of trials in which the event happened / Total number of trials...

Experimental probability is calculated based on actual experiments or trials. It helps quantify the likelihood of an event happening based on historical data from those trials. However, practical limitations arise when experimentation is expensive or difficult, necessitating theoretical probability.

Suppose you want to find the probability of getting a tiny bit of chocolate in a cookie. You could bake several batches (trials) and see how often you get chocolate versus how often you don't and use that to find your experimental probability.

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.

Theoretical probability calculates the likelihood of an event based on the assumption of equally likely outcomes. This is especially useful for events that can be theoretically assessed rather than practically tested.

Think about a standard deck of cards. The theoretical probability of drawing an Ace can be calculated by knowing there are 4 Aces in a 52-card deck. You don’t need to draw multiple cards to know the probability; it can be deduced mathematically.

The probability of an event which is impossible to occur is 0... The probability of an event which is sure (or certain) to occur is 1.

In probability theory, events are categorized into certain events (which will definitely happen) and impossible events (which cannot happen). This classification helps categorize events effectively within the probability framework.

If you roll a six-sided die, the probability of rolling a 7 is impossible, so the probability is 0. If you roll the die, you will definitely get a number from 1 to 6, so that probability is 1.

Example 1: Find the probability of getting a head when a coin is tossed once... Simultaneously, if F is the event ‘getting a tail’, then P(F) = 1/2.

This example illustrates how to calculate simple probabilities based on the number of favorable outcomes versus possible outcomes. It reinforces the concept with specific numerical calculations.

Imagine again flipping a coin. The chance of landing on heads is simply 1 out of the 2 possible outcomes. Likewise, the chance of tails is the same, and this equality helps us understand the fairness of the coin.

An event having only one outcome of the experiment is called an elementary event... the sum of the probabilities of all the elementary events of an experiment is 1.

Elementary events represent the simplest forms of probable outcomes. Understanding these foundational events provides insights into calculating overall probabilities for more complex events, emphasizing that the total probability must always add up to 1.

Consider each individual flip of a coin (getting heads or tails) as an elementary event. When you analyze a series of flips, even though the outcomes vary, the probabilities must still adhere to the principle that all possible outcomes collectively contribute to a total of 100% certainty.

For any event E, P(E) + P(not E) = 1... E and E are called complementary events.

Complementary events encompass all possible outcomes of an event. Understanding these pairs helps simplify calculations in probability, since knowing the probability of one event automatically provides the probability of its complement.

If you consider passing or failing an exam as complementary events, if you know the probability of passing (say 0.75), you effortlessly determine that the probability of failing must be 0.25.

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