# Probability

*How likely something is to happen.*

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.

### Tossing a Coin

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

Heads (H) or

Tails (T)

Also:

- the probability of the coin landing
**H**is ½ - 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 \frac{1}{6}

## Probability

In general:

Probability of an event happening = \frac{Number of ways it can happen}{Total number of outcomes}

### Example: the chances of rolling a "4" with a die

**Number of ways it can happen: 1** (there is only 1 face with a "4" on it)

**Total number of outcomes: 6** (there are 6 faces altogether)

So the probability = \frac{1}{6}

### Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

**Number of ways it can happen: 4** (there are 4 blues)

**Total number of outcomes: 5** (there are 5 marbles in total)

So the probability = \frac{4}{5} = 0.8

## Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

## Probability is Just a Guide

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

### Example: 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.

Learn more at Probability Index.

## Words

Some words have special meaning in Probability:

**Experiment**: a repeatable procedure with a set of possible results.

### Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

**Outcome:** A possible result.

### Example: "6" is one of the outcomes of a throw of a die.

**Trial:** A single performance of an experiment.

### Example: I conducted a coin toss experiment. After 4 trials I got these results:

Outcome |
Trial | Trial | Trial | Trial |
---|---|---|---|---|

Head | ✔ | ✔ | ✔ | |

Tail | ✔ |

Three trials had the outcome "Head", and one trial had the outcome "Tail"

**Sample Space:** all the possible outcomes of an experiment.

### Example: 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: Deck of Cards

- the 5 of Clubs is a sample point
- the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 *different* sample points.

### Example: Throwing dice

There are 6 different sample points in that sample space.

**Event:** one **or more** outcomes of an experiment

### Example Events:

An event can be just one outcome:

- Getting a Tail when tossing a coin
- Rolling a "5"

An event can include more than one outcome:

- Choosing a "King" from a deck of cards (any of the 4 Kings)
- Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

### Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The **Sample Space** is all possible **Outcomes** (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ...

...

... {6,3} {6,4} {6,5} {6,6}

The **Event** Alex is looking for is a "double", where both dice have the same number. It is made up of these **6 Sample Points**:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

Trial | Is it a Double? |
---|---|

{3,4} | No |

{5,1} | No |

{2,2} | Yes |

{6,3} | No |

... | ... |

After 100 **Trials**, Alex has 19 "double" **Events** ... is that close to what you would expect?