How likely something is to happen.

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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)

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 16

Probability

In general:

Probability of an event happening = Number of ways it can happenTotal 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 = 16

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 = 45 = 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}

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:

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 the sample space.

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a '5'

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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:

After 100 Experiments, Alex has 19 'double' Events ... is that close to what you would expect?

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