2.1 Simple Probability

Probability can be defined as the chance of an event occurring by calculating the number of desired outcomes with the total number of possibilities on a scale of 0 to 1, where a value of 1 means that the probability is very accurate and vice versa. The formula is:

\[ P = \frac{Total\,Number\,Of\,Favourable\,Outcomes}{Total\,Number\,Of\,Possible\,outcomes} \] The video provides an example of tossing a coin once and calculating the probability of getting Heads or Tails. This can be calculated as follows:

Looking for Head Opportunities \[ P(H)=\frac{Head}{Head\,and\,Tails} \vee \frac{1}{2} \] Looking for Tails Opportunities: \[ P(T)=\frac{Tails}{Tails\,and\,Head} \vee \frac{1}{2} \] So it can be concluded that the probability of getting heads and tails is 0.5, or 50% when converted to a percentage. Then, if heads appear twice, then: \[ P(HH)=0,5\times0,5=0,25 \] or in percentage terms, it can be converted to 25%, so that the probability of getting heads twice is 25% and vice versa.

2.2 Sample Space

Sample space is the set of all possible outcomes of an experiment or random event. For example, when tossing a coin, the possible outcomes are “heads” or “tails,” so the sample space = {heads, tails}. If you toss the coin twice, the sample space = {0,1,2} because there is a chance of getting 0 heads, 1 head, and 2 heads.

• Tossing one coin can result in Heads or Tails and has a value of: \[ P(H)=0,5 \vee P(T)=0,5 \]
• The second coin toss can also result in Heads or Tails from the first toss and has a value of \[ P(H)=0,25 \vee P(T)=0,25 \]

So if we create Outcomes, the toss of 2 coins becomes:

• HH
• HT
• TH
• TT

From these outcomes, we know that there are four possible outcomes when tossing a coin twice. To determine the probability of the coin, we can multiply the probabilities of each event together. \[ P=0,5\times0,5=0,25 \] Therefore, the probability of getting Heads or Tails on the second coin toss is 25%.

2.3 Complement Rule

The complement rule states that the probability of an event not occurring is 1 minus the probability of that event occurring. For example, if the probability of getting two tails when tossing a coin twice is 0.25, then the probability of not getting two tails is: \[ P=1-0,25=0,75 \] This rule is useful for calculating the probability of complex events or makes it easier to calculate the probability of complex events. So we can see that:

• Probability values are between 0 and 1 \[ 0<P(A)<1 \]
• The total sum of all possible probabilities is 1. \[ P(A)'=1-P(A) \] Conclusion: It can be seen that probability requires a clear definition of sample space and the complement rule to calculate odds.

3.1 Independent Events

Independent Events are when two events occur and one event does not affect the probability of the other event occurring. Examples include rolling a die or flipping a coin; the results of both are not related and do not influence each other.

The formula is: \[ P(A\,and\,B)=P(A)\times P(B) \] This means that the probability of both events occurring simultaneously is the result of multiplying the probabilities of each event.

Example of Independent Events: Throwing a die and a coin. Suppose you want to get a 5 on the die and get heads on the coin. Then the probability of getting a 5 on the die is 1/6 and the probability of getting heads on the coin is 1/2. So the probability of both is: \[ \frac{1}{6}\times\frac{1}{2}=\frac{1}{12} \]

3.2 Dependent Events

Dependent events occur when two or more events happen within a single event and one event influences the other. An example of this is when taking balls from a box without returning them; after the first ball is taken, the composition changes, thereby influencing the probability of the next event.

Formula: \[ P(A\,and\,B)=P(A)\times P(B|A) \] P(B|A) means the probability of event B occurring after event A occurs. This formula reflects the change in probability because the first event affects the second event.

Example of Dependent Events: A child takes two marbles from a box containing 7 green and 3 blue marbles without returning them. The probability of drawing a green marble first is 7/10, and the probability of drawing a blue marble second is 3/9.

Therefore, the probability of both events occurring simultaneously is: \[ \frac{7}{10}\times \frac{3}{9}=\frac{7}{30} \] Conclusion: It can be seen that the video clearly explains the difference between independent and dependent events in probability, and tells us how to calculate the probability for each type of event.

4.1 Review of Basic Probability Definitions

This video reviews the material from the basic probability definitions, which have the following formula: \[ P = \frac{Total\,Number\,Of\,Favourable\,Outcomes}{Total\,Number\,Of\,Possible\,outcomes} \] and also discusses sample space, which refers to all possible outcomes of an experiment. To understand the material on the union of events, you must understand the definition of probability in order to better understand the material on the union of events. So, the important points to know are:

• Sample space is the set of all outcomes of an experiment
• The probability of an event is the desired outcome divided by the total number of outcomes.

4.2 Union of Events

The video explains the concept of combined probability (UNION) of two events occurring and the use of sample space and Venn diagrams to understand the concept easily. For example, if there are events A and B, this can be expressed as A union B or written as (A ∪ B). The video gives an example of throwing two dice, which has a sample space of 36 possible outcomes. Thus, the probability of an event is defined as

The video provides an example of rolling two dice, which has a sample space of 36 possible outcomes. Thus, the probability of an event is defined as the sum of the desired outcomes divided by the total number of possible outcomes in the sample space.

The formula is: \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] When P(A ∩ B) is the probability of events A and B occurring simultaneously.

Conclusion: It can be seen that using the Union of Events formula is very important to avoid double counting overlapping results, and Venn diagrams help with visualization by making it easier to understand the concept.

5.1 Mutually Exclusive Events

Two events can be called mutually exclusive if they cannot occur simultaneously. For example, when throwing a die, the numbers 3 and 5 cannot appear at the same time. Therefore, the probability of both events occurring is zero: \[ P(A\cap B)=0 \] Since there is no overlap between the two events, the formula for the combined probability of two mutually exclusive events is: \[ P(A\cup B)=P(A)+P(B) \]

5.2 Exhaustive Events

An event can be said to be complete if the entire combination of events covers all possible outcomes in the sample space. This means that no outcome is excluded from any event. For example, when throwing a single die, the events of an odd number and an even number appearing are complete events because they cover all possible outcomes.

Unlike mutually exclusive events, if both events are complete, then the combined probability of both events is 1. \[ P(A\cup B)=1 \]

However, mutually exclusive events are not always mutually exclusive because events can overlap but still cover the entire sample space.

Conclusion: The video explains that mutually exclusive events cannot occur simultaneously, while complementary events cover all possible outcomes in the sample space. If events are mutually exclusive and collectively exhaustive, their combined probability can be calculated simply, and the total probability is = 1. This is important for accurate probability analysis and when calculating a probability using a Venn diagram for visualization.

6.1 Binomial Experiments

There are four conditions in binomial experiments, namely:

• The number of experiments (n) is fixed
• Each experiment has only two outcomes, namely success or failure
• The probability of success (p) is fixed for each experiment conducted
• Each experiment conducted is independent (the result of one experiment does not affect the results of others)

An example of a binomial experiment is when tossing a coin three times. Each toss can result in heads or tails, but the probability remains 0.5, and each toss does not affect the others.

Binomial formula: \[ P(X=k)= \binom{n}{k}\cdot P^{k}\cdot (1-p)^{n-k} \] Explanation:

• (Combination) calculates the number of ways to select k successful trials from n trials. It can be written as: \[ \binom{n}{k} \]
• The probability of success occurring in k trials. Can be written as: \[ p^{k} \]
• The probability of failure occurring in the remaining trials can be written as: \[ (1-p)^{n-k} \]

When is it applied? Many real-life situations are suitable for modeling as binomial experiments. For example, when repeatedly tossing a coin, checking for defects in a series of products, counting how many students pass an exam, and so on. If you don’t recognize the situation as binomial, it is difficult to calculate the probability of the “number of successes” accurately; while by understanding the binomial distribution, we can predict more systematically.

Conclusion: The video provides a complete explanation of how binomial experiments and binomial formulas work in statistics. It explains that there are four important conditions in a binomial experiment, and the binomial formula allows us to calculate the probability of success from a number of trials. The binomial distribution is very useful in data analysis and probability prediction in many fields: economics, social research, quality control, health, and so on.