A Probability shows us the likelihood of something happening.

Probability is assigned a positive number that lies between 0 and 1.

If we are 100% sure that something will happen, then the probability is 1. Conversely, if it’s completely impossible for something to happen, then the probability is 0.

Probability can be expressed as a:

• Number that’s between 0 and 1 e.g. 0.6, 0.8, 0.364, etc.
• Percentage e.g. 1%, 78%, 99%, etc.
• Fraction, e.g., ½, ¼, ¾, etc.

The higher the probability, the more likely that the defined event will occur. The smaller the probability, the more unlikely that something will happen.

Example:
If the probability of Chelsea FC winning the Premier League is 56% and the probability of Arsenal FC winning the Premier League is 39%, then Chelsea FC is more likely to win than Arsenal FC.

Probability is used to describe uncertain events

Examples:
When a coin is tossed, it may either land with the heads facing upward or with the tails facing upwards.
When you roll a dice, the number of dots that appear on top could be anything from one dot to six dots.

## Terminologies Used in Probability

From the two examples above, we can define the following terminologies used in probability.

1. ### Experiment

This is the uncertain process in question

e.g.

• coin tossing
• dice rolling
2. ### Outcome

This is the final result after the experiment has occurred.

e.g.

• Heads appearing after a coin has been tossed
• Three dots coming up after a dice has been rolled
3. ### Sample space

This is a set of all the outcomes that are possible for a particular event.

E.g.

• A tossed coin may land in any of the two ways; heads or tails.
• A cast dice may present 1,2,3,4,5, or 6 dots.

A symbol S is used to denote the sample space

The symbol n(S) is used to indicate the size of the sample space, i.e. the complete number of outcomes that are possible.

Examples

• A tossed coin may land with either the tails (T) or heads (H). Therefore, the sample space is S= {H; T} and the size of the sample space is n(S)= 2
• A cast dice may land with any number of dots from 1 to 6 facing upwards. Therefore, the sample space is S {1;2;3;4;5;6} while the size of the sample space is n(S)= 6
4. ### Event

This is a specific outcome of an experiment. We are usually interested in the event and not the other possible outcomes

A symbol E is used to show an event

A symbol n(E) is used to show the size of an event. The size of an event is the number of outcomes in that event

Examples

• A coin is tossed, and the head comes up. The single outcome is the event, and it will be expressed as E={H}. The size of the outcome will be n(E) = 1
• A dice is cast, and three dots show up. The single outcome is the event, and it will be expressed as E= {3}. The size of the event will be n(E) = 1

## Theoretical Probability

Probability is just a theoretical ratio between the number of specific outcomes and number of all possible outcomes i.e.

Therefore:

In Summary:
n(E) is the number of specific outcomes (events) in which we are interested.
n(S) is the total number of possible outcomes.
p(E) is the probability.

Example 1

### Let’s calculate the probability of a coin landing on tails after it’s tossed

n(E)= 1
n(S)= 2

Probability P(E)= ½
= 0.5
= 50%

### What’s the probability of three dots showing up after a dice is cast?

n(E)= 1
n(S)= 6

Probability P(E)= 1/6
= 0.167
= 16.7%

Example 2

A young woman picks a beauty magazine that has 100 pages. She reads one page from the magazine. What is the probability that the page that she chose had an even page number?

Solution:

There are 50 even page numbers in that magazine. Therefore, n(E) = 50

The young woman could have opened any one of the 100 pages. Therefore, the total number of possible outcomes is 100 and thus, n(S) = 100

Probability p(E) = 50/100
= 50%
= 1/2
= 0.5

Therefore, the possibility that the page has an even page number can be expressed as 0.5 (a number), ½ (a fraction) or 50% (a percentage)

Example 3

A basket has 6 toys. 3 of those toys are green, 2 are blue, and 1 is yellow. A child picks a random toy from the basket. Determine the probability that the toy is green.

Solution:

There are 3 green toys hence n(E) = 3

The child could have picked any of the 6 toys hence n(S) = 6

Probability p(E) = 3/6
= ½
= 0.5
= 50%
Therefore, the possibility that the child picks a green toy can be expressed as 0.5 (a number), ½ (a fraction) or 50% (a percentage)

Similarly probability for picking up blue toy is 1/3 (0.33 or 33.33%) and probability of picking up yellow toy is 1/6 (0.66 or 66.66%).

Example 4

What is the probability of a 6-sided dice landing on either 3 or 5?

We want the dice to land on either of the two sides (3 or 5) hence n(E) = 2

However, the dice can land on any of its 6 sides hence n(S) = 6

Probability p(E)= 2/6
= 1/3
= 0.33
= 33.33%

Therefore, the possibility of the dice landing on either 3 or 5 can be expressed as 0.33 (a number), 1/3 (a fraction) or 33% (a percentage)