8.01 Theoretical probability as likelihood
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:
We say that flipping the coin is a trial, and there are two equally likely outcomes: heads and tails. The list of all possible outcomes of a trial is called the sample space.
Another example of a trial is rolling a die:
There are 6 equally likely outcomes in the sample space: 1, 2, 3, 4, 5, and 6. We can group these outcomes into events, such as "rolling an even number" or "rolling more than 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:
| Likehood | Event |
|---|---|
| \text{Impossible} | \text{Rolling a}\ 9 |
| \text{Unlikely} | \text{Rolling a}\ 1 |
| \text{Even chance} | \text{Rolling }\ 4\ \text{or more} |
| \text{Likely} | \text{Rolling }\ 2\ \text{or more} |
| \text{Certain} | \text{Rolling between}\ 1\ \text{and} \ 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, or that each outcome is equally likely.
If every outcome corresponds to the event, we say it is certain. If no outcomes correspond to the event, we say it is impossible.
Let's look at another example. This is a full set of 52 playing cards:
There are many different possible events, depending on what result we are interested in:
Two colors: Red and Black
Four suits: Spades, Hearts, Clubs, Diamonds
Thirteen card values:
Three "face cards": "K" for King, "Q" for Queen, "J" for Jack
The numbers 2 through 10
"A" for "Ace", which is usually given the value of 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:
| Likehood | Event |
|---|---|
| \text{Impossible} | \text{Drawing a}\ 17 \text{ of Hearts} \\ \text{Drawing a blue card} \\ \text{Drawing a } 2 \text{ of Clubs} |
| \text{Unlikely} | \text{Drawing an Ace} \\ \text{Drawing a face card} \\ \text{Drawing a Spade} |
| \text{Even chance} | \text{Drawing a black card } \\ \text{Drawing a red card} |
| \text{Likely} | \text{Drawing a card number }\ 2\ \text{through } 10 \\ \text{Drawing a card that is not }\ 2 \\ \text{Drawing a card of any suit that is not Hearts} |
| \text{Certain} | \text{Drawing a card that is a Spade, Heart, Club, or Diamond }\\ \text{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 lesson.
Examples
Example 1
A six-sided die is rolled in a trial. What are the chances that the outcome is 2 or more?
Example 2
Look at this spinner:
What is the most likely symbol to spin?
What is the likelihood of spinning a ball?
Trial - a single experiment with different possible results.
Outcome - the possible results of a trial.
Event - a grouping of outcomes. Each possible outcome is always an event on its own.
Likelihood - an event can be:
impossible - can never happen
unlikely - happens less than half the time
even chance - happens half the time
likely - happens more than half the time
certain - always happens
Sample space - a list of all the possible outcomes of a trial.