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Sure, maybe, never

Probability · level 1, lesson 1 of 6

A question to start

Here are three things. Think about each one.

  1. The sun will set tonight.
  2. It will snow tomorrow.
  3. A cat will drive you to school in a car.

Will they happen?

The first one will happen. You do not need to wait and see. The sun sets every evening, so you can be sure.

The third one will not happen. Cats cannot drive. It will never happen, however long you wait.

The second one is the interesting one. Maybe it will snow, and maybe it will not. You cannot be sure until tomorrow comes. Most of the things in life are like this: they might happen.

These three little words, sure, maybe and never, are where the math of chance begins. Grown-ups call this math probability. It is the math of "how likely is it?"

The idea

Three piles

Imagine a pile of cards. On each card is something that could happen. Your job is to sort the cards into three piles.

suremaybenever the sun willset tonight a dropped ballfalls down it will snowtomorrow a friendvisits today a cat drivesa car a dog fliesto the moon
  • Sure means it will happen. There is no other way it can turn out. Another word for sure is certain.
  • Never means it cannot happen. There is no way at all for it to turn out like that. Another word for never is impossible.
  • Maybe means it might happen and it might not. Both ways are possible. You have to wait and see.

Every card goes in exactly one pile. A thing cannot be sure and never at the same time.

How do you know it is sure?

To put a card in the sure pile, you need a reason. Here are some good reasons.

  • It always happens. The sun has set every evening for as long as there have been people to watch it. The Earth keeps turning, so it will set tonight too.
  • There is nothing else it can be. If you drop a ball, it falls down. It cannot fall up.
  • It is a rule. Tomorrow will come after today. Monday comes after Sunday. That is just how days work.

A good test for "sure" is to ask: "Can I think of any way that this does not happen?" If you cannot think of any way at all, it is sure.

How do you know it is never?

To put a card in the never pile, you also need a reason.

  • It cannot be done. Cats cannot drive. Dogs cannot fly to the moon by flapping their ears.
  • It is not one of the choices. If a bag has only blue blocks in it, you can never pull out an orange block. There is no orange block in there to pull out.

A good test for "never" is to ask: "Can I think of any way that this does happen?" If you cannot think of any way at all, it is never.

Everything else is maybe

If you can think of a way that it happens, and a way that it does not happen, it is maybe.

  • Will it snow tomorrow? It could, if the clouds are cold enough. It could not, if the sky is clear. Maybe.
  • Will a friend visit today? They could. They could stay home. Maybe.
  • Will you find a coin on the path? Maybe.

Most of the interesting questions in the world are maybes. That is why probability is useful: it helps us think about maybes.

A bag of blocks makes it easy to see

Bags of blocks are a great way to practise, because you can look inside and see every possible block.

bag Abag Bbag C

Close your eyes, put your hand in, and pull out one block.

  • From bag A, you will surely get blue. Every block in there is blue. Orange is never.
  • From bag B, you will maybe get blue and maybe get orange. Both are in there.
  • From bag C, blue is never, and orange is sure.

Look how the same question, "Will I get blue?", has three different answers. It depends on what is in the bag. That is a big idea: whether something is sure, maybe or never depends on what could happen.

It can depend on where you are

"It will snow tomorrow" is a maybe in winter in Canada. But in a hot desert in summer, it is so close to never that nobody would bet on it.

And "it will rain tomorrow" is a maybe almost everywhere on Earth. But on Mars, it is a never: the air there is so thin and so cold that water cannot fall as rain. The same words can go in a different pile in a different place.

So when you sort, think about where and when the thing would happen. Sometimes the right answer is "it depends", and saying what it depends on is a very good answer.

Sure is not the same as "I think so"

Here is a tricky one. "My mum will make dinner tonight." That feels sure. She makes dinner every night! But could something be different tonight? Maybe the family goes out to eat. Maybe a grandparent cooks. So really it is a maybe, a very likely maybe.

Mathematicians are careful with the word "sure". They only use it when there is no way at all for the thing not to happen. "Very likely" is still maybe. You will learn about "likely" in the next lesson.

Worked examples

Example 1. Sort six cards.

The cards are:

  • A: "You will have a birthday this year."
  • B: "You will roll a 7 on an ordinary die." (A die is one of a pair of dice. It has 1, 2, 3, 4, 5 or 6 dots on its faces.)
  • C: "Your teacher will wear a hat today."
  • D: "Tomorrow will come after today."
  • E: "A fish will climb a tree."
  • F: "You will get a 6 when you roll a die."

Go one card at a time and ask both tests.

  1. A: You have a birthday every year. No way it does not happen. Sure.
  2. B: A die has only 1 to 6 dots. There is no face with 7 dots. No way it happens. Never.
  3. C: Your teacher might wear a hat, and might not. Maybe.
  4. D: That is how days work. Sure.
  5. E: Fish have no legs and live in water. Never.
  6. F: The die might land on 6. It might land on 1, 2, 3, 4 or 5. Maybe.

Piles: sure = A, D. Maybe = C, F. Never = B, E. Check: 2 + 2 + 2 = 6 cards. None lost.

Example 2. One bag, three questions.

A bag has 3 green blocks and nothing else. You pull out one block without looking.

  1. "It will be green." Every block is green. Sure.
  2. "It will be blue." There is no blue block in the bag. Never.
  3. "It will be a block." Everything in the bag is a block. Sure.

Now someone drops 1 blue block into the bag. The bag has 3 green and 1 blue.

  1. "It will be green." Now it could be blue. Maybe.
  2. "It will be blue." It could be. Maybe.
  3. "It will be a block." Still sure.

Adding one block moved two answers from sure and never to maybe.

Example 3. Rolling a die.

Roll one ordinary die.

  1. "It will show a number from 1 to 6." Every face has 1 to 6 dots. Sure.
  2. "It will show 0." No face is blank. Never.
  3. "It will show 3." It could, and it could not. Maybe.
  4. "It will show a number smaller than 10." Every face is smaller than 10. Sure.

Notice number 4. It does not sound sure at first, but when you check every face, all six of them are smaller than 10. Checking every possible way is how you know.

Try it

  1. Sure, maybe or never? "Water will be wet."
  2. Sure, maybe or never? "Your shoe will turn into a frog."
  3. Sure, maybe or never? "You will see a bird today."
  4. A bag has 5 orange blocks and nothing else. You pull out one block. Is getting orange sure, maybe or never? Is getting blue sure, maybe or never?
  5. A bag has 2 orange blocks and 2 blue blocks. Is getting orange sure, maybe or never?
  6. Roll a die. Sure, maybe or never? (a) It shows 8. (b) It shows a number bigger than 0. (c) It shows 5.
  7. Make a bag of blocks where getting blue is never and getting green is a maybe.
  8. Sam says: "My team will win the game tomorrow. We always win, so it is sure." Is Sam right?
Answers
  1. Sure. Water is always wet. There is no way it is not.
  2. Never. Shoes cannot turn into frogs.
  3. Maybe. You probably will, but you might not (say, if you stay inside all day). This is a "very likely maybe", not sure.
  4. Getting orange is sure: every block is orange. Getting blue is never: there is no blue block.
  5. Maybe. It could be orange, and it could be blue.
  6. (a) Never: no face has 8 dots. (b) Sure: every face has at least 1 dot. (c) Maybe: it could be 5, and it could be another number.
  7. There are many right answers. You need no blue blocks, some green blocks, and something else too, so that green is not sure. For example: 2 green and 3 orange. Blue is never (no blue), and green is maybe (it could be green or orange).
  8. No. Even a team that always wins could lose one day: a player is ill, or the other team plays very well. Winning is likely, but it is a maybe, not sure.

Common mistake

Putting very likely things in the sure pile. "I will see a bird today" feels sure, but there is a way it does not happen. Sure means no way at all for it not to happen. If you can think of even one way, it is a maybe.

Putting very unlikely things in the never pile. "I will find a lost toy today" might feel like never, but it could happen. Never means no way at all for it to happen. A very small chance is still maybe.

Where it leads