An Introduction to Elementary Probability

Probability is the likelihood of happening of something with respect to the total number of outcomes. For example, soon or later, most of us have asked ourselves the following questions:

1. What is the likelihood that it is going to rain tomorrow?
2. What is the likelihood that when flipping a coin, it will land on tail?
3. What is the likelihood that a patient will die after a surgery?

The second question above is easier to answer.

Since a coin has two sides, it has 50% chance of landing on tail. It also has 50% chance of landing on head. The first and the third question are not easy to answer since they may require a solid understanding of the topic.

My goal is to introduce you to the topic and help you develop an appreciation towards it. Any serious study starts with a solid understanding of the fundamental counting principle, combination, and arrangements.

Fundamental Counting Principle

We will introduce the fundamental counting principle with an example. This counting principle is all about choices we make when there are different possibilities.

Suppose most of your clothes are dirty and you are left with 2 trousers and 3 shirts. How many choices do you have or in how many different ways can you dress? Let’s call the pair of trousers: trousers 1 and trousers 2. Lets call the shirts: shirt 1, shirt 2, and shirt 3. So you can try trouser 1 with shirt 1 or shirt 2 or shirt 3, so there are 3 possibilities, or three cases. In the same way, if you take trouser 2, you can wear it with shirt 1 or shirt 2 or shirt 3 and again there are three possibilities or three different cases.

Now you can’t wear both the trousers at the same time, hence that cannot be considered an option.

Hence the total number of possibilities or cases is 6 (3+3)

We can also say, total number of possibilities is equal to , because there are two different trousers and three different shirts

In general, if you have choices for a first task and choices for a second task, you have choices for both tasks

Another example:

You go a restaurant to get some breakfast. The menu says pancakes, waffles, or home fries and in beverages you can choose from coffee, juice, hot chocolate, tea. How many different choices of food and beverage do you have?

There 3 choices for food and 4 choices for beverage.

Thus, you have a total of choices.

Theoretical Probability

The theoretical probability is found whenever you make use of a formula to find the probability of an event. To find the probability of an event, also called likelihood of an event, use the formula below:

Probability of an event

The number of favorable outcomes is the likelihood to get a specific outcome.

For example, suppose you throw a die numbered from 1 to 6. Count all the possible numbers you can get. This is called number of possible outcomes. All the possible numbers are 1, 2, 3, 4, 5, and 6. Thus the number of possible outcomes is 6.

You could make up different types of favorable outcomes

You could say…

1. Likelihood to get an even number
2. Likelihood to get a prime number
3. Likelihood to get an odd number
4. Likelihood to get a 4.
5. Likelihood to get a 1.
6. Likelihood to get a number bigger than 4
7. Likelihood to get a number less than 6

All the above are favorable outcomes.

Example: Throw a die once. What is the probability of getting a number less than 6?

Ask yourself, “How many number are less than 6?”

Since there are 5 numbers less than 6, the number of favorable outcomes is 5

Since the die had a total of 6 numbers, the number of possible outcomes is 6. Probability of getting a number less than 6

Probability of getting a number less than 6

Probability of getting a number less than 6 = 0.8333

Probability of getting a number less than 6 = 83.33%

Probability of getting a yellow ball = 0.3333

This means that it is very likely you will get a number less than 6

Example: A bag contains 6 blue balls, 4 yellow balls, and 2 red balls. What is the theoretical probability of getting a yellow ball?

Since you have 4 yellow balls playing on your favor, the number of favorable outcomes is 4.

To get the number of possible outcomes, just count all the balls.

Number of possible outcomes is 12

Probability of getting a yellow ball 

Probability of getting a yellow ball = 0.3333

Probability of getting a yellow ball = 33.33%

Probability of Compound Events

The probability of compound events combines at least two simple events. The probability, that a coin will show head when you toss only one coin, is a simple event. However, if you toss two coins, the probability of getting 2 heads is a compound event because once again it combines two simple events.

Suppose you say to a friend, “I will give you 10 dollars if both coins land on head.”

Let’s see what happens when your friend toss two coins:

The different outcomes are HH, HT, TH, or TT.

As you can see, out of 4 possibilities, only 1 will give you HH.

the probability of getting 2 heads is 

Your friend has 25% chance of getting 10 dollars since one-fourth = 25%.

The example above is a good example of independent events.

What are independent events?

When the outcome of one event does not affect the outcome of another event, the two events are said to be independent. In our example above, when you toss two coins, neither coin has the power to influence the other coin. This compound event is independent then. When two events are independent, you can use the following formula:

.

Let’s use this formula to find the probability of getting 2 heads when two coins are tossed

Coin 1:

Probability of getting head 

Coin 2:

Probability of getting another head

Probability (Heads and Heads)

Probability (Heads and Heads)