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More likely and less likely

Probability · level 1, lesson 2 of 6

A question to start

Here is a bag. Inside are 5 orange blocks and 1 blue block. All six blocks are the same size and feel the same.

You shake the bag, close your eyes and pull out one block. Before you look, make a guess: will it be orange or blue?

Most people guess orange. They are right to. But why? And if you guess orange, will you always be right? Can the blue block ever come out?

The idea

Both are maybe

First, sort it the way we did in the last lesson. Is getting orange sure? No: the blue block is in there, so you might get it. Is getting blue never? No: it is in there too. So both are maybe.

But they are not the same kind of maybe. One of them happens much more often than the other. To talk about that, we need two new words.

  • More likely means it happens more often.
  • Less likely means it happens less often.

Getting orange is more likely than getting blue. Getting blue is less likely than getting orange.

Why orange is more likely: count the ways

Here is the reason. When you shake the bag and grab without looking, every block has the same chance of being the one you grab. They are all the same size. They all feel the same. Your hand cannot tell them apart. So no block is special.

Now give each block a number, so you can talk about it.

123 456 6 blocks, each just as easy to grab

Your hand will grab one of these six. There are 6 ways it can turn out: block 1, block 2, block 3, block 4, block 5 or block 6.

  • How many of those ways give orange? Blocks 1, 2, 3, 4 and 5. That is 5 ways.
  • How many give blue? Only block 6. That is 1 way.

Orange has 5 ways and blue has 1 way. The colour with more ways is more likely. That is the whole secret.

Will you always be right?

No. "More likely" does not mean "sure". The blue block really is in the bag, and sometimes your hand will find it. If you guess orange every time, you will be right most of the time, but not every time.

That can feel unfair. You made the smart guess and you were still wrong! But that is what maybe means. A good guess is one that is right most often, not one that is right every time.

Trying it

You can test this at home with any small things that feel the same: blocks, buttons, coloured paper squares folded the same way, or socks. You need 5 of one colour and 1 of another.

  1. Put them in a bag or a box you cannot see into.
  2. Shake it. Pull one out without looking.
  3. Look at the colour and make a mark on paper.
  4. Put it back. This is important: if you keep it out, the bag changes.
  5. Do it again, 12 times in all.

To keep count, use tally marks. Each time, draw one line. When you get to five, draw the fifth line across the other four. That makes counting easy: you can count the bundles in fives.

Here is what one family got. Your tally will be different, and that is fine.

orange: 10blue: 2

Orange came out 10 times and blue 2 times. Orange won by a lot, just as the counting said. Blue did come out, twice. "Less likely" is not "never".

If you do it again, you might get 11 and 1, or 9 and 3, or even 12 and 0. The numbers jump around each time. But orange nearly always wins. If you count, you will see why: in every 6 grabs, you would expect about 5 orange and 1 blue, so in 12 grabs, about 10 orange and 2 blue. "About" is the right word. It is not exact.

Same number means same chance

What if the bag has 3 orange and 3 blue?

Now orange has 3 ways and blue has 3 ways. Neither colour has more ways. They are equally likely: each one has the same chance. If you guess orange, you are right about half the time. If you guess blue, the same.

More is not the whole story

Here is a puzzle for careful thinkers. Which bag is better if you want orange?

bag Pbag Q

Bag Q has 4 orange blocks. Bag P has only 2. So bag Q is better? No!

Look at each bag on its own.

  • In bag P, orange has 2 ways and blue has 1 way. Orange is more likely than blue.
  • In bag Q, orange has 4 ways and blue has 4 ways. They are equally likely.

So bag P is the better bag for orange. What matters is not just how many orange blocks there are, but how many orange blocks there are compared with the blue ones. Bag Q has more orange, but it also has a lot more blue.

What if the blocks are not the same?

All of this works only if your hand cannot tell the blocks apart. Suppose the blue block is huge and the orange ones are tiny. Your hand would bump into the big blue block first, nearly every time! Then "count the ways" would give the wrong answer.

So for a fair test, the things in the bag must be the same size, the same shape and the same feel. Only the colour is different, and you cannot feel colour.

Worked examples

Example 1. Which colour is more likely?

A bag has 2 green blocks and 7 orange blocks. You grab one without looking.

  1. Every block is just as easy to grab.
  2. Green has 2 ways. Orange has 7 ways.
  3. 7 is more than 2, so orange is more likely. Green is less likely.
  4. Green is still possible. It is a maybe, not a never.

Example 2. Three colours.

A bag has 4 orange, 1 blue and 2 green blocks.

  1. Orange has 4 ways, blue has 1 way, green has 2 ways.
  2. Put them in order, from most likely to least likely: orange, then green, then blue.
  3. The most likely colour is orange. The least likely is blue.

Example 3. Making it equal.

A bag has 5 orange and 2 blue blocks. You want orange and blue to be equally likely. What could you do?

  1. Orange has 5 ways and blue has 2 ways. They are not equal.
  2. One way: add 3 blue blocks. Then there are 5 orange and 5 blue.
  3. Another way: take out 3 orange blocks. Then there are 2 orange and 2 blue.
  4. Either way, both colours have the same number of ways, so they are equally likely.

Try it

  1. A bag has 1 orange block and 4 blue blocks. Which colour is more likely?
  2. A bag has 6 green blocks and 6 orange blocks. Which colour is more likely?
  3. A bag has 3 orange, 3 blue and 1 green. Which colour is least likely?
  4. A bag has 8 blue blocks and 1 orange block. Mia says, "I will never get orange." Is she right?
  5. A bag has 4 orange and 1 blue. You want blue to be more likely than orange. How many blue blocks could you add?
  6. Bag P has 3 orange and 1 blue. Bag Q has 5 orange and 5 blue. Which bag is better if you want orange?
  7. You pull a block out of a bag 12 times, putting it back each time. You get orange 11 times and blue once. Make a guess: does the bag have more orange or more blue?
Answers
  1. Blue. Blue has 4 ways and orange has 1 way.
  2. Neither. They have 6 ways each, so they are equally likely.
  3. Green. It has only 1 way. Orange and blue have 3 ways each.
  4. No. Orange is less likely (1 way against 8), but the orange block is in the bag, so it can come out. It is a maybe, not a never.
  5. Blue needs more ways than orange, so more than 4 blue blocks. There is 1 already, so add 4 or more. Adding 4 gives 5 blue and 4 orange. (Adding exactly 3 gives 4 and 4: equally likely, not more.)
  6. Bag P. In bag P, orange has 3 ways and blue has 1, so orange is more likely. In bag Q they are equal, 5 and 5. Bag Q has more orange blocks, but it has just as many blue ones too.
  7. More orange, probably many more. If the bag had the same number of each, you would expect about 6 and 6. Getting 11 orange is a strong hint that orange has many more ways. You cannot be sure without looking, but it is a good guess. You will do more detective work like this in level 2.

Common mistake

Thinking "more likely" means "it will happen". Orange is more likely from a bag of 5 orange and 1 blue, but you will sometimes get blue. More likely means more often, not every time.

Looking only at one colour. A bag with 10 orange blocks sounds great for orange. But if it also has 20 blue blocks, orange is less likely. Always compare the colours with each other.

Where it leads