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.
-
Experiment
This is the uncertain process in question
e.g.
- coin tossing
- dice rolling
-
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
-
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
-
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.
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%
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)
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%).
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)
