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# Basic Probability

We describe probabilities in our everyday lives. For example, you might say that "it is likely to rain later", "I am probably not going to finish my homework" or "there is an even chance of heads or tails". These all involve probability buzzwords, such as 'probably', 'likely', 'certain', 'impossible', and many more.

The next time you watch the news, see how many of these buzzwords you can spot. There will probably be a lot!

In this article, we will recap the basic probability theory, we'll see the basic probability formula and examples of its application.

## Probability theory basics: a recap

Using mathematical theory, we can numerically describe or estimate the probability of an event occurring, instead of just using words.

What is an event?

An event is a possible outcome that results from a random experiment. It is usually denoted with a capital letter, like A, B, C, etc. An event is regarded as a subset of a sample space in probability.

Then, what is a sample space?

A sample space is a set that contains all likely events or outcomes of an experiment. It is usually represented with the Greek letter Ω (omega) or with S.

If an event is certain to occur (so it is definitely going to happen), then it has a probability of 1.

An example of a certain event is the probability that tomorrow is Tuesday given that it is currently Monday. This will always be the case, therefore the chances of it happening are 100%.

If an outcome is impossible (so it is definitely not going to happen), then it has a probability of 0.

An example of an impossible event is the probability that tomorrow is Tuesday given that it is currently Thursday. This will never happen, therefore the chances of it happening are zero.

Say you flip an unbiased coin. There will be the same chance of seeing either 'heads' or 'tails'. Since the likelihoods are the same, there must be a probability of 0.5 that 'heads' is flipped. Similarly, there is also a 0.5 probability of 'tails' being flipped.

Do you notice anything from these examples? The probability of something occurring is always between 0 and 1, which represent impossible and certain events respectively. There can never be a negative probability, and the chances of something occurring can never be larger than 1. In most cases, events occurring will be neither certain nor impossible, in which case the probability can be represented using either a fraction (e.g. ), a decimal (e.g. ), or a percentage (e.g. ).

A probability can never be greater than 1, or 100%, or less than 0, or 0%.

## Calculating probabilities: the basics

There are two main types of probabilities that you need to be aware of: theoretical probability and experimental probability. Theoretical probability is determined using reasoning and the theoretical knowledge of a particular situation. Experimental probability uses the outcomes of an experiment to infer the probability of an event occurring in the future.

However, there is a third type of probability referred to as axiomatic probability but it is out of the scope of this article; you need not concern yourself with this as it is an advanced topic.

Here, we will use theoretical probability to analyse the properties of a six-sided die. We will also introduce the basic probability formula.

### The basic probability formula

Say you rolled an unbiased, 6-sided die. What is the probability that you roll a 3?

Solution

There are 6 sides and therefore there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6. This is our sample space. We can notate this as S = {1, 2, 3, 4, 5, 6}.

Since the die are unbiased, we know that these outcomes are all equally likely. The likelihood of rolling a 3, therefore, is .

Indeed, since all the outcomes are equally likely, the probability of rolling any particular number is .

We can generalise this example using the following definition.

The probability of an event A occurring is:

.

Using this in our die example: the total number of possible outcomes is 6 and there is only 1 possible way to roll a three. We can therefore write

.

### The probability of an event not happening

If we know the probability of an event occurring, we can also work out the probability of it not happening.

Notice that if we wanted to work out the probability of rolling a number between 1 and 6, we can write the following:

.

Intuitively, the event rolling between 1 and 6 is a certainty since these are the only options available.

The fact that all probabilities must add up to 1 is useful in working out the probability of something not happening. For example, the probability of not rolling a three is the same as the probability of rolling either 1, 2, 4, 5, 6, which, using the formula, is

.

Alternatively, we could have subtracted the probability of failure from 1:

.

The probability of an event, A, not happening is:

.

Using set notation:

.

What is the probability of rolling a 1, 2, 3, or 6?

Solution

Since this is equivalent to asking what the probability of not rolling a 4 or 5, we can do the following:

Therefore, using the formula :

## Examples of calculating basic probabilities

To better understand how to calculate basic probabilities, more examples should be practiced.

A fair die and a coin were rolled once. What is the probability of;

a) having an even number.

b) having either an even or a prime number.

Solution

a) having an even number;

To find the probability of an even number, We know we have three even numbers in a fair die, that is 2, 4, and 6. Therefore, the probability of an even number becomes;

b) having either an even or a prime number;

In question a), we found the probability of finding an even number as;

Next, we need to find the probability of a prime number occurring. We know we have three prime numbers in a fair die, that is 2, 3, and 5. Therefore, the probability of a prime number becomes;

Therefore the probability of an even or a prime number is;

## Basic Probability - Key takeaways

• Probabilities fall between 0 and 1
• A certain event has a probability of 1 and will always happen
• An impossible event has a probability of 0 and will never happen
• Most probabilities fall somewhere between 0 and 1
• Theoretical probability is determined using reasoning and the theoretical knowledge of a particular situation
• Experimental probability uses the outcomes of an experiment to infer the probability of an event occurring in the future
• The probability of an event A occurring is
• The probability of an event A not occurring is , or using set notation

Basic probability can be used to work out the likelihood of simple events such as a single die roll or a coin flip.

The basic probability formula is the number of ways for an outcome to occur divided by the total number of outcomes.

The three types of probability are theoretical probability, experimental probability and axiomatic probability.

An example of basic probability is a coin flip: what is the probability of 'heads'? Since the probability of heads or tails is equal, the probability of heads is a half, or 0.5.

We can use mathematics and logic to solve basic probability problems.

## Final Basic Probability Quiz

Question

Which of the following are true?

Probabilities are numbers between 0 and 10

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Question

What is the probability of a certain event?

1

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Question

What is the probability of an impossible event?

0

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Question

What is the difference between theoretical and experimental probability?

Theoretical probability is determined using reasoning and the theoretical knowledge of a particular situation. Experimental probability uses the outcomes of an experiment to infer the probability of an event occurring in the future

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Question

What is the probability of an event not happening?

The probability of an event not happening is 1 minus the probability of the event happening.

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Question

Name three ways to describe probabilities.

Using a fraction (e.g. 1/2), a decimal (e.g. 0.5) or a percentage (e.g. 50%)

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Question

What is independent event probability?

Independent event probability is when the occurrence of one event does not influence the probability of another event happening.

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Question

The probability of two independent events happening at the same time is the same as the intersection of both events. True or False

True

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Question

You roll a standard 6-sided dice. How many ways can you roll an odd number?

There are 3 odd numbers on a dice, so there are 3 ways to roll an odd number.

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Question

A lottery machine contains balls with numbers from 1 to 10. All the numbers come out just as often as one another. What is the probability that the next ball drawn is a 1?

10%

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Question

I have an unfair coin. When I toss this coin, the probability of getting heads is 0.2. What is the probability of getting tails?

0.8

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Question

I have a deck of 54 playing cards: 52 regular cards and 2 Jokers. I shuffle the cards thoroughly. What is the probability that the first card I draw is a Joker?

2/52

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Question

Scarthorpe City and Scarthorpe United are rival football teams. They have played many matches before. Sometimes City wins, sometimes United wins. The one thing the experts can agree on is that the outcome of the next match is very hard to predict.

If the experts know what they're talking about, which of these could be the probability that Scarthorpe United will win?

0

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Question

An internet marketer knows from past performance that the probability of a spam email leading to a sale is 0.001%. They send out 100 million emails. If performance remains the same, how many sales will this lead to?

10

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Question

I am not a good archer. I shot an arrow 2000 times, and it hit the target just 20 times. What is the experimental probability that I hit the target?  a) 20% b) 10% c) 0.1 d) 0.01

20%

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Question

What does 'mutually exclusive' mean?

Two events are mutually exclusive if they cannot happen at the same time.

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Question

Are the following events mutually exclusive?

Rolling a 6 and rolling an even number

Yes

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Question

Are the following events mutually exclusive?

Drawing a 4 from a deck of cards, and drawing a diamond.

Yes

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Question

Are the days of the week mutually exclusive?

Yes

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Question

Fashions change, but at the time of writing, everyone agrees that neckties should be worn with shirts, not t-shirts. A necktie would look ridiculous with a t-shirt.

Based on the above, which one of the following is true about wearing a t-shirt and wearing a necktie at the time of writing?

They are mutually exclusive events.

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Question

I'm planning what I will do this evening. I could go to a restaurant, I could cook a meal at home, I could go to the cinema and I could go to the theatre. I won't eat two meals. I can't go to both the cinema and the theatre, because their shows are at the same time.

Which of the following pairs of events must be mutually exclusive?

Going to the cinema and going to the theatre

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Question

According to a travel guide, the country of Bhutan has no traffic lights, but there are traffic wardens everywhere.

In Bhutan, are traffic wardens and traffic lights mutually exclusive?

Yes. Since there are no traffic lights, the probability of traffic lights is 0. Since traffic wardens are everywhere, the probability of traffic wardens is 1. Since these two cannot happen at the same time, they are mutually exclusive events.

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