12.01 The language of probability

Probability is the study of chance and prediction. To make sure our predictions are valid, we need to use the right mathematical language.

In general we will be thinking about a single test, known as a trial (also known as an experiment), that has more than one possible result, known as an outcome. A good example is flipping a coin.

Another example of a trial is rolling a die:We say that flipping the coin is a trial, and there are two equally likely outcomes: head, and tails. The list of all possible outcomes of a trial is called the sample space.

A single die	All possible faces

There are $6$6 equally likely outcomes in the sample space: $1$1, $2$2, $3$3, $4$4, $5$5, and $6$6. We can group these outcomes into events, such as "rolling an even number" or "rolling more than $3$3". Each outcome on its own is always an event, and sometimes events don't correspond to any outcomes.

We can think about different kinds of events that we care about, and sort them into categories of likelihood. Here are some examples when rolling a die:

Likelihood	Event
Impossible	Rolling a $9$9
Unlikely	Rolling a $1$1
Even chance	Rolling $4$4 or more
Likely	Rolling a $2$2 or more
Certain	Rolling between $1$1 and $6$6

What makes an event likely or unlikely depends on what happens when you repeat the trial many times. If the event happens more than half the time, we say it is likely, and if it happens less than half the time, we say it is unlikely. If it happens exactly half the time we say it has an even chance.

If every outcome corresponds to the event, we say it is certain. If no outcomes correspond to the event, we say it is impossible.

 

Exploration

This is a full set of $52$52 playing cards:

Notice that there are many different events, depending on what result we are interested in:

• Two colours
  • Red
  • Black
• Four suits
  • Spades  
  • Hearts 
  • Clubs 
  • Diamonds 
• Thirteen card values
  • The three "face cards"
    • "K" for "King" 
    • "Q" for "Queen" 
    • "J" for "Jack"
  • The numbers $2$2 through $10$10
  • "A" for "Ace", which is usually given the value of $1$1.

The deck of cards is shuffled, and the trial is going to be drawing a single card from the deck.

Here are some events sorted into each of the five likelihood categories:

Likelihood	Event
Impossible	Drawing a $17$17 of Hearts Drawing a blue card Drawing a $2$2 of Cups
Unlikely	Drawing an Ace Drawing a "face card" Drawing a Spade
Even chance	Drawing a black card Drawing a red card
Likely	Drawing a card numbered $2$2 through $10$10 Drawing a card that is not a $2$2 Drawing a card of any suit that is not Hearts
Certain	Drawing a card that is a Spade, Heart, Club, or Diamond Drawing a card that is either red or black

Drawing a "face card" is unlikely because there are fewer of them than the other cards. Drawing a black card has an even chance because there are just as many black cards as red cards. There are many more possible events we could describe, and fitting them into the right likelihood category can take some practice. We will investigate ways we can be precise in the next chapter.

Practice questions

Question 1

Question 2

Question 3