Tossing a coin, inspecting a product, and similar things are often repeated around us. We know that the possible consequences of these things are either 'head' or 'tail', 'good' or 'defective', but we don't know what the outcome will be. An experiment or observation in which similar things are repeated and all possible outcomes are known, but one outcome is determined by chance is called the trial.

The set of all possible outcomes that can occur in a trial is called the sample space, and a subset of this sample space is called the event. An event made up of one element is called the basic event.

The sample space is usually denoted by \(S\), and events are denoted by uppercase English letters, \(A, B, C, ...\), etc. For example, in a coin tossing trial, when 'head' is H and 'tail' is T, the sample space is as follows:

Events of 'head' and 'tail' are denoted as follows:

With respect to two events \(A\) and \(B\) in the sample space \(S\), the event in which \(A\) or \(B\) occurs is denoted by \(A \cup B\), and the event in which \(A\) and \(B\) occurs simultaneously denoted by \(A \cap B\).

If event \(A\) and \(B\) do not occur simultaneously which is \(A \cap B\)= ø, two events are called mutually exclusive.

When there is an event \(A\), the event in which \(A\) does not occur is called the complementary event of \(A\) and denoted by \(A^C\). Since \(A\) and \(A^C\) cannot occur at the same time, that is \(A \cap A^C\)= ø, the events \(A\) and \(A^C\) are mutually exclusive.

An example of finding the relative frequency by actually tossing a coin 10, 50, or 100 times and writing down the number of heads is as follows:

When the number of tosses is small, the relative frequency of the number of heads may not be 0.5. However, as the number of tosses of a coin increases, the relative frequency approaches 0.5. If you experiment more with the number of tossing a coin using a computer, it can be seen that the relative frequency gradually approaches 0.5 as shown in <Figure 2.4>. This relative frequency is called the statistical probability of an event that a coin comes up head.

In general, the probability of an event \(A\), denoted as \(P(A)\), is expressed as a numerical value between 0 and 1. Let \(n_A\) be the number of times an event \(A\) occurs when the trial is repeated \(n\) times. As the number of trials increases sufficiently, if the relative frequency \(\frac{n_A}{n}\) approaches a constant value \(p\), this value is called the statistical probability of the event \(A\) which is denoted as \(P(A) = \frac{n_A}{n}\).

It is not easy to define the probabilities of all events using statistical probabilities and use them for decision-making in reality. In order to apply the probability easily to the real problem, a mathematical probability based on the model was studied.

In order to obtain the statistical probability of each side of the dice, numerous trials of rolling the dice are required. The statistical probability obtained through sufficient trials will be close to 1/6. However, it is not easy in reality to do many trials to find the probability of one event.

If one dice is rolled, the sample space is {1, 2, 3, 4, 5, 6}. Assume that each element in the sample space has the same likelihood of occurrence. The probability of each element can be said to be 1/6. A mathematical probability is defined as the likelihood of an event under the rational assumption that each element in the sample space is equally likely to appear. When making decisions about real problems, this mathematical probability is used a lot.

In general, assuming that each element of the sample space \(S\) in a trial has the same probability of occurrence, the mathematical probability that an event \(A\) will occur denoted as \(P(A)\) is defined as follows:

In general, the statistical probability \(p\) of an event \(A\) approaches to the mathematical probability if the number of trials is large enough.

Let's examine properties of the mathematical probability when each element in the sample space has the same probability of occurrence. Since an event \(A\) is a subset of the sample space \(S\), \(n(S)\) should be greater than or equal to \(n(A)\) which should be greater than 0 as follows:

If we divide each term by \(n(S)\), it becomes as follows:

Therefore,

If \(A\) is equal to the sample space \(S\), then

If \(A\) is an empty set ∅, then

If the event where a student watched the soccer game is \(A\) and the volleyball is \(B\), the event of watching either the soccer or volleyball is \(A \cup B\). Denote the number of element in each event of \(A\), \(B\) and \(A \cup B\) as \(n(A)\), \(n(B)\) and \(n(A \cup B)\) respectively, then they satisfy the following relation.

If we divide both sides by \(n(S)\), it becomes as follows:

Therefore, the probability that either the event \(A\) or event \(B\) occurs, denoted as \(P(A \cup B)\), is as follows:

This is called the addition rule of probability. In the above example, it becomes as follows:

If two events \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B)\)= 0 and the addition rule becomes as follows:

The sample space of rolling a dice is \(S\) = {1, 2, 3, 4, 5, 6}. If the event in which the number of dots appeared is 1 or 2 is \(A\) = {1, 2}, the event of remaining numbers 3, 4, 5, 6 is called the complementary event and is denoted as \(A^C\) = {3, 4, 5, 6}. Since the event \(A\) and the complementary event \(A^C\) are mutually exclusive, the addition rule becomes as follows:

Since \(P(A \cup A^C ) = P(S) = 1\), The probability of the complementary event \(P(A^C )\) becomes as follows:

The number of events given in this problem can be summarized in the following table by denoting the event of male student as \(A\), the event of female student as \(F\), the event of wearing glasses as \(B\), and the event of not wearing glasses as \(N\).

The above table makes it easy to calculate the number of events not given by the problem as follows:

1) If you randomly select one of the high school students, the probability of being a male student is as follows:

2) If you randomly select a student, the probability that this student is a male student and wears glasses is as follows:

3) When one of the male students is randomly selected, the probability that this student wears glasses is denoted as a symbol \(P(B|A)\), and, since the event of a male student \(M\) can be viewed as a sample space, it is calculated as follows:

\(P(B|A)\) is called the conditional probability of wearing glasses (\(B\)) when a male student is selected (\(A\)).

If \(S\) is a sample space of a statistical experiment, the conditional probability of the event \(B\) when the event \(A\) occured is as follows:

If we divide both numerator and denominator by \(n(S)\), the conditional probability becomes as follows:

In the table above, the probabilities of two events occurring at the same time are summarized as follows: This is called a joint probability table.

In this table, the probability \(P(A \cap B) = \frac{8}{40}\) can be calculated by multiplying the probability of male students \(P(A) = \frac{24}{40}\) with the conditional probability \( P(B|A) = \frac{8}{24} \).

It is called the multiplication rule of probability. This formula can be derived if you multiply \(P(A)\) on both side of the definition of the conditional probability \( P(B|A) = \frac{P(A \cap B)}{P(A)} \).

The probability \(P(A \cap B) = \frac{8}{40}\) can be calculated by multiplying the probability of wearing glasses \(P(B) = \frac{12}{40}\) with the conditional probability \( P(A|B) = \frac{8}{12} \).

In general, the multiplication rule of probability is as follows:

Suppose that the joint probability table on gender of students and wearing glasses is as follows:

Here, the probability of all students wearing glasses \(P(B) = \frac{20}{40} = \frac{1}{2}\) is the same as that of male students \( P(B|A) = \frac{12}{24} = \frac{1}{2} \). If the occurrence of an event \(A\) does not affect the probability that the event \(B\) will occur, that is,

it is said that two events \(A\) and \(B\) are independent of each other. On the other hand, if two events \(A\) and \(B\) are not independent, it is said two events are dependent.

If two events \(A\) and \(B\) are independent, the rule of multiplication becomes as follows:

Inversely, if \( P(A \cap B) = P(A) × P(B) \) and \(P(A) \ne 0\), the conditional probability becomes as follows:

Therefore, two events \(A\) and \(B\) are independent.

In the table above, all events for the joint probability are independent of each other. In this case, it is said that the gender variable and the variable of wearing glasses are independent. Looking at the table, the probabilities of "wearing glasses" and "not wearing glasses" events of all students are 0.5 and 0.5, respectively. When the two variables are independent, these ratios are maintained for each male and female student.