RD Sharma Class 11 Solutions Chapter 33 - Probability (Ex 33.4) Exercise 33.4

Probability is a fun and interesting branch of mathematics that many students enjoy. It is concerned with the probability that an event will occur or not. For example, we may assign to the event 'A person falls into a pit' a probability of 0.2 (which is equal to 0.2), and we may assign to the event 'A person does not fall into a pit' a probability of 0.8 (which is equal to 0.8). A mathematical function is used to express a probability.

As another example, we may want to know the probability of rolling a particular number on dice or the probability that our car will break down today. For the first example, we need to define what we mean by a probability of 0.2 (or 0.8). We have the probability of rolling a certain number on the dice is a function of the number on the dice and is given by:

0.2 = Pr(E)

where:

E is the event (of rolling a number that has a certain value)

Pr(E) is the probability of the event A occurring (in this case it is 0.2)

Probability is related to the law of large numbers. When the number of events is sufficiently large, the law of large numbers provides a good approximation of the actual probability. The term was first used by Charles de Laplace to describe his belief that he could predict the outcomes of gambling games.

The theory of probability provides information about a random event that involves a large number of possible outcomes. The basic concept of probability theory is the probability function, which is a function from a set of outcomes, called the sample space, to the interval between 0 and 1. 

For example, if a die has six faces numbered 1, 2, 3, 4, 5, and 6, the sample space is {1, 2, 3, 4, 5, and 6}. The event that the die faces a 6 is the event E, whose probability is defined to be. More generally, if X is any event that is defined to have a probability less than or equal to p, the probability of X is denoted p.

The sample space must be specified to calculate the probability of an event. The most fundamental concept in probability is the probability space. An event is a set of outcomes. An outcome is a member of the sample space. The space can be defined in a variety of ways, depending on the probability model being used. 

A probability space for a classical probability model consists of a sample space and a probability measure on the sample space. A model other than the classical probability model may be defined by a countably additive set of events, known as a μ-space, on a probability space.

Two properties that make probability theory useful are the law of large numbers and the continuity principle. The law of large numbers states that if there are sufficiently many trials, then the probability that a given event will occur will tend toward the probability that the event will occur. For example, if there is one face on a die with six faces, then the probability that the face is 1, 2, 3, 4, 5, or 6 is   (the decimal places can be changed to accommodate another model of probability). 

However, if there are six rolls of the die, then the probability that the face is 1, 2, 3, 4, 5, or 6 is about .618, which is. The law of large numbers leads us to believe that the probability that the face is any one of the numbers 1 through 6 is approximately, but this belief is only correct if the sample space consists of and the probability of an event is defined by the event probability model for a classical probability model. 

If the sample space consists of and the probability of an event is defined by the event probability model for a discrete probability model, then the probability that the face is 1, 2, 3, 4, 5, or 6 is. The law of large numbers also leads us to believe that the probability that the face is either 1 or 2 is approximately, or This is only approximately true for a sample space consisting of and a discrete probability model for a classical probability model. In a continuous probability model, the law of large numbers does not apply. This does not mean that the model is wrong. It only means that we have to look at the data, not at the sample space, when using the probability model.