In mathematics, we study probability. Probability is so familiar in our lives and all statistics are based on probability.

It is because we study probability that we can understand what the news is about. Also, experimental data in science is processed using probabilities. Whether or not it will rain in a weather forecast is also a probability. If you don’t know what probability means, you won’t be able to understand the data correctly and you will get into a lot of trouble when you go through your daily life as an adult.

That’s why we study probability in mathematics. Mathematics must be learned in order to use it in everyday life, and probability is a type of mathematics. When learning the basics of probability in mathematics, the most frequently used examples are coin and dice probabilities.

What are the probabilities when we toss two or more coins or dice? We will explain the basics of probability as taught in mathematics.

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## What Is Probability? Numbers of Rates That Occur

First, what is probability? Probability is the rate at which a particular event occurs. For example, what is the probability that a 4 will come up when we roll the dice? For this, the probability can be answered by anyone as $\displaystyle\frac{1}{6}$.

When rolling the dice, if we roll the dice six times, we can expect to get a 4 at some point. Of course, we might roll the dice and get a 4 on the first time. In some cases, you might finally get a 4 on the 20th time. But in any case, the probability of coming up with a 4 on the dice is $\displaystyle\frac{1}{6}$.

So how do we get the probability? In the case of dice, there are numbers from 1 to 6. That is, there are six different ways to come up with the numbers. Out of the six, only one of them can be number 4. Therefore, the probability is $\displaystyle\frac{1}{6}$.

Hence, in probability, the following formula holds.

In calculating the probability, be sure to count the number of times of the total event. With dice, there are six different results. Out of them, we can put the number of times a particular event will occur in the numerator to get the probability.

For example, what is the probability of getting an even number on the dice? These must be understood because probability frequently raises the question of even and odd numbers.

As mentioned earlier, the dice have between 1 and 6 possible outcomes. Of these, 2, 4 and 6 are even numbers. That is, there are three possible ways to get an even number. So the probability of getting an even number on the dice is $\displaystyle\frac{3}{6}$. However, the answer is $\displaystyle\frac{1}{2}$, since we need to reduce a fraction.

We can use the same method to come up with probabilities for odd numbers. When calculating probabilities in mathematics, try to think about how many ways there are in total, and how many times the particular event will occur.

### Equally Possible Means Same Frequency of Occurrence

However, when calculating the probability, there are assumptions. That assumption is called equally possible. What does it mean to be equally possible?

When we roll the dice, the probability of getting 4 is $\displaystyle\frac{1}{6}$, as mentioned above. Then, is the probability of coming up with a 4 for the following dice also $\displaystyle\frac{1}{6}$?

The probability of coming up with 4 is considerably higher than $\displaystyle\frac{1}{6}$. Why is the probability of getting a 4 not $\displaystyle\frac{1}{6}$, despite the dice? The reason for this is that it is biased to be a specific number by the shape of dice.

The calculation of probability requires the assumption that the probability of occurrence is the same for all events. This is equally possible.

For example, when betting, in many cases the challenger loses. One reason for this is that cheating takes place. In betting, there are few cases that are equally possible. If it is not equally possible, we cannot calculate the probability.

## Calculating the Probability of the Front and Back of Two Coins

Let’s actually calculate the probability. In calculating probability, we will consider a situation in which two coins are tossed. When we learn the basics of probability in mathematics, we often use the coin example to learn probability.

A coin has two sides; face and reverse. The probability of tossing a coin and getting a face or a reverse is $\displaystyle\frac{1}{2}$. So what is the probability if we toss two coins?

When calculating the probability, if we think intuitively, there is a high probability of being incorrect. Therefore, we have to understand the correct way to get the probability. For example, what is the probability that the two coins will both be on the front? Those who get it wrong answer $\displaystyle\frac{1}{3}$. This is because we think there are the following types.

- Front, Front
- Front, Back
- Back, Back

So, let’s think about it more precisely. For example, suppose that there are the following two coins (5 and 10).

Toss two of these coins and we can understand that there are four ways.

There are several (front, back). There are two types: (front, back) and (back, front). Therefore, the total number of events to occur is four. Therefore, the probability of tossing two coins and having them both face up is $\displaystyle\frac{1}{4}$.

### Find the Probability by Tree Diagram

However, it is difficult to write down all the events without a hint. How can we find all the events efficiently? One way to do this is with a tree diagram. By using a tree diagram, we can write out all events.

In a tree diagram, we focus on one event at first. When we focus on a single coin, we get either front or back. Therefore, we can draw a tree diagram as follows.

Then toss the other coin. As a result, we can see that there are four ways as follows.

In other words, a tree diagram is a way to write out all the different types of events. Whether we toss a coin at the same time or separately, the results of the coin will remain the same. So use the tree diagram to check all events.

### Not Drawing a Tree Diagram Is Likely to Be Incorrect

As mentioned earlier, many people make miscalculations when calculating probabilities by intuition. In other words, if we don’t draw a tree diagram, we make a lot of miscalculations.

The reason why miscalculation occurs when tossing two coins is that many people think about the same type coin tossing situation. What is the probability that both coins will be on the front side when tossing the same two coins, as shown below?

Even if they are the same coin, the way of thinking is the same as before. Even if the coins are the same or different, the probability is the same. Just because the coins are the same does not change the probability.

In this case, since the coins are of the same type, we need to think of the tree diagram with different criteria. So, let’s divide them up by the first coin and the second coin. Although the two coins are actually tossed at the same time, we assume that they are tossed separately for the purpose of drawing the tree diagram. Then we have the following.

When drawing a tree diagram, it is important to draw events as they are assumed to occur separately, rather than at the same time. As a result, we can draw all events.

Then, check the number of times the event is happening from the tree diagram. For example, when tossing two coins, it would be as follows.

- Both front: one
- One side front, the other side back: two
- Both back: one

By drawing a tree diagram, we can find out the number of times of the total event. Not only that, we can count how many times a particular event has occurred. When learning the basics of probability in mathematics, be sure to draw a tree diagram.

## The Probability of Rolling Two Dice

After understanding the probability of coins, let’s learn about the probability of tossing two dice. In the basics of probability, the dice are a frequently asked question.

Whether the two dice are the same or different sizes, the result is the same. We have explained that the idea of probability is the same regardless of whether the coins are the same or different. If we draw a tree diagram, the answer will be the same for both. The same can be said for dice, not coins.

So, how should we think of the tree diagram for two dice? First of all, in the case of the first roll of the dice, there are six different ways. Then, when we roll the dice for the second time, there are six different ways for each.

In fact, we roll the dice at the same time. However, as explained so far, we make a tree diagram assuming that the dice are rolled separately. In that case, we get the following.

The first dice has 6 ways, and the second dice has 6 ways. Therefore, there are $6×6=36$ events.

However, in the case of dice, it is difficult to distinguish them because of the complexity of the tree diagram. Also, it takes time to create a tree diagram. So, when rolling two dice at the same time, it is common in mathematics to create the following table.

It is not available if we roll three dice. However, this table is useful in the case of rolling two dice.

For example, if we roll two dice, what is the probability that the multiplying numbers will come to 12? If we want to find this probability, check the box where multiplication will result in 12. It will look like the following.

In the table, there are four ways in which the result will be 12. The overall event is 36, so the probability of rolling two dice and getting the result to be 12 is $\displaystyle\frac{4}{36}$. And by reducing a fraction, the answer is $\displaystyle\frac{1}{9}$.

### Calculate the Probability of Not Occurring

In mathematics probability, sometimes we have to calculate the probability of not happening. For the probability of occurrence, we can calculate it according to the method described so far. On the other hand, how can we calculate the probability of non-occurrence?

The idea is the same when calculating the probability of non-occurrence. First, try to find the probability of occurrence.

For example, suppose there is a lottery that has a 20% chance of winning. In this case, what is the probability of being a loser in the lottery? The calculation is as follows.

- 100% – 20% = 80%

There are only two types of lotteries: winning and losing. Therefore, by subtracting the probability of winning from 100%, we get the probability of being a loser (the probability of not happening).

Of course, the idea is the same for fractions instead of percentages. 20% has the same meaning as $\displaystyle\frac{1}{5}$. That is, the probability of being a winner is $\displaystyle\frac{1}{5}$. In this case, the probability of being a loser is as follows.

- $1-\displaystyle\frac{1}{5}=\displaystyle\frac{4}{5}$

100% = 1. Once we understand these facts, we can determine the probability of not happening by the following formula.

- Probability of not happening $=1-$ probability of happening

In this way, we can find the probability that it will not happen. Understand not only the probability of happening, but also how to get the probability of not happening.

## Exercises: Probability Calculation Problems

**Q1:** Solve the following problem.

Roll two dice. After squaring the number of dice, if adding the two numbers, find the probability that the number of dice will be less than or equal to 30.

**A1:** Answer.

In dice problems, be sure to make a table. After making the table, squared each number. Then check the part that adds up to 30 or less. It will look like this.

The overall event is 36. Also, the number of checks is 19. Therefore, we have the following probability.

- $\displaystyle\frac{19}{36}$

**Q2:** Solve the following problem

There are three envelopes with numbers 1 to 3 written on them. Also, there are three cards with the numbers 1 to 3 written on them.

Then put all the cards into each of the envelopes. What is the probability that the numbers on the envelopes and the cards are all different?

**A2:** Answer.

Without drawing a tree diagram, we can’t solve the problem. So, let’s draw a tree diagram.

What if we put card 1 in envelope 1? In this case, we can put either card 2 or 3 in envelope 2. Then, we would put the remaining cards in Envelope 3.

In the same way, we will draw a tree diagram, considering the case of putting card 2 in envelope 1 and the case of putting card 3 in envelope 1. Then we will have the following.

Thus, we can see that there are six ways in total. In which of these cases are the envelope and card numbers all different? Counting them one by one, there are two such events as follows.

- Envelope 1 (card 2), envelope 2 (card 3), envelope 3 (card 1)
- Envelope 1 (card 3), envelope 2 (card 1), envelope 3 (card 2)

Therefore, the probability is as follows.

- $\displaystyle\frac{2}{6}=\displaystyle\frac{1}{3}$

## Calculating the Basis of Probability

When we learn the basics of probability in math, we don’t start out learning difficult formulas. By using a tree diagram, we can figure out what patterns are there, one by one. Therefore, without drawing a diagram, we cannot come up with an answer.

If we solve a problem without a tree diagram, we are almost certain to get an incorrect answer. This is because, as explained in the coin example, without a diagram, mistakes occur frequently.

Probability problems are more challenging because all patterns need to be shown. Not only that, but we have to count how many target events are occurring.

By doing these things one by one, we will know what the probability of occurrence is. At the same time, we will be able to calculate the probability that it will not happen. When learning probability, understand these basic concepts.