# Probability Intro, Probability of Flipping a Coinand Rolling a Dice, Notation

Probability is a measure of how likely an event or outcome is to possibly occur.

### Values of Probability:

The value of a probability can have a value between &nbsp0&nbsp and &nbsp1.

Where probabilities with a value nearer to &nbsp0&nbsp being less likely to happen,
and probabilities with a value nearer to &nbsp1&nbsp being more likely to happen.

So an event having a probability of &nbsp0&nbsp means it can’t occur at all.

Where as an outcome having a probability of &nbsp1&nbsp means that it will definitely occur.

It’s handy to demonstrate this using a what we can call a &nbsp”probability number line”,&nbsp shown below.

### How to Work out Probability

The Probability of a certain event can be established as a fraction sum.

Event Probability &nbsp&nbsp = &nbsp&nbsp\frac{\tt{\color{blue}Number \space of \space ways \space an \space event \space can \space happen}}{\tt{\color{red}Number \space of \space different \space possible \space outcomes}}

The amount that is the number of different possible outcomes, is referred to as as the &nbsp”SAMPLE SPACE“.

So the sample space in probability is the set of all the outcomes that are possible with a certain event.

## Working out Probability Examples

### Probability of Flipping a Coin

A good way to get introduced to probability is looking at the probability of flipping a coin.

We can write the probability of this event as a fraction sum.

The possible outcomes from flipping the coin, are a &nbsp”HEAD”&nbsp or a &nbsp”TAILS”.

SAMPLE SPACE &nbsp=&nbsp { HEAD , TAILS }

Let’s look at the probability of landing a head.

There is only &nbsp1&nbsp way that this event can happen,
and there are &nbsp2&nbsp possible outcomes from flipping a coin, we can land a &nbspHEAD&nbsp or a &nbspTAILS.

Probability of a HEAD &nbsp&nbsp = &nbsp&nbsp \frac{\tt{\color{blue}Number \space of \space ways \space an \space event \space can \space happen}}{\tt{\color{red}Number \space of \space different \space possible \space outcomes}}

Probability of a HEAD &nbsp = &nbsp \bf{\frac{1}{2}}

It should be mentioned that it is also the case that probabilities can be written in the form of decimals.

The probability of landing a &nbspHEAD&nbsp was &nbsp{\frac{1}{2}}.

Which is a fraction that can be written as &nbsp0.5.
Thus as a percentage, there is a &nbsp50%&nbsp probability of landing a &nbspHEAD&nbsp with the flip of a fair coin.

In this case there are &nbsp2&nbsp possible ways that this can occur, and &nbsp2&nbsp possible outcomes in total.

Probability of a HEAD or a TAIL &nbsp = &nbsp \bf{\frac{2}{2}} &nbsp=&nbsp 1

A probability value of &nbsp1&nbsp means that the result from a coin flip will definitely be either a &nbspHEAD&nbsp or a &nbspTAILS.

Percentage wise a &nbsp100%&nbsp probability.

### Probability of Rolling a Dice

An example that expands further on the probability of flipping a coin, is the event of rolling a six standard sided dice, which is another good example to introduce probability in Math.

The sides of a standard dice are comprised of &nbsp6&nbsp different numbers.

SAMPLE SPACE &nbsp=&nbsp { 1 , 2 , 3 , 4 , 5 , 6 }.

Each of these numbers has the same probability of being rolled, &nbsp \bf{\frac{1}{6}}.

Probability of Rolling an Odd Number:
There are &nbsp3&nbsp sides of the dice with an odd number,&nbsp 1 , 3 , 5.

Probability of rolling an odd number &nbsp=&nbsp \bf{\frac{3}{6}} &nbsp=&nbsp \bf{\frac{1}{2}} &nbsp=&nbsp 0.5 &nbsp=&nbsp 50%

Probability of Rolling a 4:
There is just one side of the dice with a number &nbsp4.

Probability of rolling a 4 &nbsp=&nbsp \bf{\frac{1}{6}} &nbsp=&nbsp 0.1666 &nbsp=&nbsp 16.66%

Probability of Rolling a 9:
Now to think about the probability of rolling a number &nbsp9.

There turns out to be no side with the number &nbsp9&nbsp on a standard six sided dice.

So there are &nbspzero&nbsp ways that this outcome can occur, however there are still &nbspsix&nbsp possible outcomes that could occur from rolling the dice.

Probability of rolling a &nbsp9 &nbsp = &nbsp \bf{\frac{0}{6}} &nbsp=&nbsp 0

A probability of &nbsp0&nbsp for rolling a &nbsp9,&nbsp tells us that it can’t happen in any way.

There is a &nbsp0%&nbsp probability.

## Probability Notation

Probability notation is generally written with a capital &nbsp”P“,&nbsp accompanied by the relevant event that could occur.

P( Event ) &nbsp=

With the probability of flipping a coin example above, the probability of landing a HEAD was &nbsp50%.

Using appropriate notation for probability this would be written as:

When dealing with the probability more then one event, there is symbols we use for “and” or “or” situations.

The probability of events A and B is notated, &nbsp&nbsp P( AB ).

The probability of events A or B is notated, &nbsp&nbsp P( AB ).

### Probability Complement

The “probability complement”, is the probability of an event not happening.

With the rolling of a dice example above, we looked a the probability of rolling a number &nbsp4.

The probability compliment of that event is the probability of NOT rolling a number &nbsp4.

The notation for the complement can be done in one of several different ways.

Let’s say we have an event labelled simply as &nbspX,&nbsp then the complement of this event can be written as:

Xc &nbsp , &nbsp X &nbsp or &nbsp X.

The notation &nbspAc&nbsp is probably the from that will be encountered the most for the probability complement.

### Probability of the Complement:

One important thing to realize and remember is that the probability of an event happening, and the probability of the event compliment, add up to &nbsp1.

P(X) + P(Xc) &nbsp=&nbsp 1

As a consequence we get how to compute the value of the complement. &nbsp&nbsp P(Xc) &nbsp=&nbsp 1P(X)

So for the example with the rolling a dice earlier.

P( 5 ) &nbsp=&nbsp \bf{\frac{1}{6}}

Thus the complement which is NOT rolling a &nbsp5&nbsp can be worked out with the sum:

P(5c) &nbsp=&nbsp 1P(5) &nbsp = &nbsp 1\bf{\frac{1}{6}} &nbsp = &nbsp \bf{\frac{5}{6}}

### Sample Space:

The sample space which represents all the possible outcomes of an event, is denoted with a capital &nbsp”S“.

So the sample space for rolling dice would be written as, &nbsp S &nbsp=&nbsp { 1 , 2 , 3 , 4 , 5 , 6 }.

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