Mathematics: Probability

Definition of Probability

Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.

Many events can’t be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

How likely something is to happen
Probabilities are written as fractions or decimals, and less often as percentages

Keywords of Probability

Impossible
Unlikely
Even chance
Likely
Certain
Event
Outcome
Mutually exclusive
Independent
Conditional

Words above are the best describes the chance of these events happening :

It will rain tomorrow
I will wear red shirt tomorrow
A flipped coin lands on heads
A penguin will be found in the Arctic
I will eat chocolate today

Scale of Probability

Probability is measure on scale from 0 to 1
If an event is impossible, it has a probability of 0
If an event is certain, it has a probability of 1

Tossing a Coin

When a coin is tossed, there are two possible outcomes :

Heads (H) or
Tails (T)

We say that the probability of the coin landing H is ½ and the probability of the coin landing T is ½.

Throwing Dice

When a single die is thrown, there are six possible outcomes : 1, 2, 3, 4, 5, 6.

The probability of any one of them is 1/6.

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide.

Example 1

Toss a coin 100 times, how many Heads will come up?

Probability says that Heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it, we might get 48 Heads, or 55 Heads … or anything really, but in most cases it will be a number near 50.

Problem :	A spinner has 4 equal sectors colored yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner? What are the chances of landing on red?
Solution :	The chances of landing on blue are 1 in 4, or one fourth. The chances of landing on red are 1 in 4, or one fourth.

This problem asked us to find some probabilities involving a spinner.
Let’s look at some definitions and examples from the problem above.

Definition	Example
An experiment (trial) is a situation involving chance or probability that leads to results called outcomes.	In the problem above, the experiment is spinning the spinner. Other : Tossing a coin, throwing dice, etc.
An outcome is the result of a single trial of an experiment.	The possible outcomes are landing on yellow, blue, green or red.
An event is one or more outcomes of an experiment.	One event of this experiment is landing on blue.
Probability is the measure of how likely an event is	The probability of landing on blue is one fourth.

In order to measure probabilities, mathematicians have devised the
following formula for finding the probability of an event.

Sample Space

All the possible outcomes of an experiment

Example 2

Choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc.}

The Sample Space is made up of Sample Points :

Sample Point

Just one of the possible outcomes

Example 3

Deck of Cards

The 5^th of Clubs is a Sample Point
The King of Hearts id a Sample Point

“King” is not a Sample Point. As there are 4 Kings that is 4 different Sample Points.

Sample Point

A single result of an experiment

Example 4

Getting a Tail when tossing a coin is an Event
Rolling a 5 is an Event

An Event can include one or more possible outcomes :

Choosing a King from a deck of cards (any of the 4 Kings) is an Event
Rolling an Even Number (2, 4 or 6) is also an Event

In general :

The probability of event ‘A’ is the number of ways event ‘A’ can occur divided by the total number of possible outcomes. Let’s take a look at a slight modification of the problem from the top of the page.

Here are the examples with solutions :
The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

Tree Diagrams

Independent events and their probabilities can be shown on a tree diagram. Each event is represented by a branch.

Example 5

A coin is ﬂipped twice. Draw a tree diagram to show all the possible outcomes.

Outcomes Probabilities :

HH ½ x ½ = ¼
HT ½ x ½ = ¼  
TH ½ x ½ = ¼
TT ½ x ½ = ¼

Example 6

A box contains 3 red ties and 2 white ties. John picks a tie and puts it on in the morning and puts it back at night. Draw a tree diagram to show the possible outcomes over two days.