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Experiments and tallies

Probability · level 2, lesson 3 of 6

A question to start

Your friend has a bag with 10 blocks in it. Some are orange and some are blue, but you are not allowed to look inside. You may only do this: put your hand in, pull out one block, look at it, and put it back. You can do that as many times as you like.

Can you work out how many orange blocks are in the bag?

You can never be completely sure without looking. But you can get surprisingly close. This lesson shows you how to be a chance detective.

The idea

Two directions

So far you have been going in one direction. You knew what was in the bag, and you worked out what would probably come out. "5 orange and 1 blue, so orange is more likely."

Now we go the other way. You see what comes out, and you work out what is probably in the bag. This is how scientists work. They cannot look inside the world's bag. They do experiments, count what happens, and make a careful guess.

Step 1: draw many times and keep a tally

Each draw goes like this:

  1. Shake the bag well, so the blocks are mixed.
  2. Pull out one block without looking.
  3. Make a tally mark for its colour.
  4. Put it back.

Do it 30 times. Here is what one child got.

orange: 22blue: 8

Orange: 5, 10, 15, 20, 21, 22. Blue: 5, 6, 7, 8. Check: 22 + 8 = 30. ✓

Step 2: turn the tally into a share

The bag has 10 blocks, so it helps to think about the draws in groups of 10. 30 draws is 3 groups of 10.

7 orange8 orange7 orange

Share out the 22 orange draws among the 3 groups as evenly as you can: 7, 8 and 7. So in each group of 10 draws, about 7 were orange.

Step 3: make a guess, and say how sure you are

If a bag of 10 blocks gives about 7 orange in every 10 draws, the best guess is that about 7 of its 10 blocks are orange, and 3 are blue.

Why does that make sense? Think the other way round. If the bag really had 7 orange and 3 blue, then every draw would have 7 orange ways out of 10. In 30 draws, you would expect about 21 orange: 7 for each group of 10. Getting 22 is very close to 21. So the bag "7 orange, 3 blue" fits what we saw very well.

But be honest: could it be a different bag?

  • 8 orange, 2 blue? You would expect about 24 orange in 30 draws. We got 22. That is not far off. It could be this bag.
  • 6 orange, 4 blue? You would expect about 18. We got 22. Possible, but a bit of a stretch.
  • 5 orange, 5 blue? You would expect about 15. Getting 22 or more from a half-and-half bag is very rare: it happens less than 1 time in 100.
  • 10 orange, 0 blue? Impossible! We saw blue blocks. A bag with no blue could never give a blue.

So a good detective says: "Probably 7 orange. Maybe 6 or 8. Almost certainly not 5 or fewer, and definitely not 10." That is a much better answer than just "7", because it says how sure you are.

Why more draws make a better guess

Chance wobbles. You saw that in the coin flip experiment. With only a few draws, the wobble can fool you.

Suppose you drew just 3 times and got orange, orange, orange. You might guess "all orange". But a 7-and-3 bag gives three oranges in a row quite often: about 1 time in 3. Three draws are not enough to tell.

With 30 draws, the wobble is smaller compared with the total. With 100 draws, smaller still. The more you draw, the closer your share gets to the real share in the bag. That is why scientists repeat experiments many times.

Why put it back?

If you did not put the blocks back, the bag would change after each draw. After 10 draws the bag would be empty, and you would know exactly what was in it! That is fine for a bag. But in real life you often cannot empty the bag.

  • How many fish in a lake are trout? You cannot catch every fish.
  • How many people in a town like a new park? You cannot ask everyone.

So people look at a sample: a small part, chosen by chance, and use the share in the sample to guess the share in the whole. Your bag with its put-back draws is a small model of exactly that.

When you cannot count the ways

Sometimes there is nothing to count at all. Drop a paper cup. Will it land on its side, on its top, or on its bottom? The three results are not equally likely, and no counting can tell you the chances. The only way to find out is an experiment: drop it 50 times and keep a tally. Experiments give you chances that thinking alone cannot.

Worked examples

Example 1. A three-colour bag.

A bag has 10 blocks of three colours. In 40 draws (putting back each time) you get 20 orange, 12 blue and 8 green. Guess what is in the bag.

  1. Check: 20 + 12 + 8 = 40. ✓
  2. 40 draws is 4 groups of 10.
  3. Share each colour among the 4 groups: orange 20 ÷ 4 = 5, blue 12 ÷ 4 = 3, green 8 ÷ 4 = 2.
  4. Best guess: 5 orange, 3 blue, 2 green. Check: 5 + 3 + 2 = 10 blocks. ✓

Example 2. Which bag?

You know the bag is one of two: bag P has 9 orange and 1 blue; bag Q has 5 orange and 5 blue. You draw 20 times and get 11 orange and 9 blue. Which bag is it more likely to be?

  1. From bag P you would expect about 18 orange in 20 draws (9 in every 10).
  2. From bag Q you would expect about 10 orange in 20 draws (5 in every 10).
  3. We got 11. That is very close to 10 and far from 18.
  4. It is much more likely to be bag Q. (Getting 9 blues from bag P, which has only 1 blue block in 10, would be extremely unusual.)

Example 3. Not enough draws.

Ella draws 4 times from a 10-block bag and gets 2 orange and 2 blue. She says, "The bag is exactly 5 and 5."

  1. 2 out of 4 is half, so 5 and 5 fits.
  2. But a 4-and-6 bag or a 6-and-4 bag could easily give 2 and 2 in four draws. Even a 3-and-7 bag could.
  3. Four draws are far too few to be sure. Ella should say "maybe about half", and then draw many more times.

Try it

  1. A tally of 20 draws from a 10-block bag shows orange 14, blue 6. What is the best guess for the bag?
  2. In 50 draws from a 10-block bag you get 10 green and 40 orange. Guess the bag.
  3. You draw 30 times from a bag and never get blue. Can you be sure there is no blue block in it?
  4. You draw 30 times and get 1 blue. Can you be sure there is a blue block in it?
  5. Why do you have to put each block back before the next draw?
  6. Bag P has 2 orange and 8 blue. Bag Q has 8 orange and 2 blue. You draw 10 times and get 7 orange. Which bag is it probably?
  7. Max drew 10 times and got 6 orange. Mia drew 100 times from the same bag and got 71 orange. Whose guess about the bag should you trust more, and why?
Answers
  1. 20 draws is 2 groups of 10. Orange: 14 ÷ 2 = 7 per 10. Best guess: 7 orange and 3 blue.
  2. 50 draws is 5 groups of 10. Green: 10 ÷ 5 = 2. Orange: 40 ÷ 5 = 8. Guess: 2 green and 8 orange.
  3. No. If there were 1 blue block in 10, you would probably see it in 30 draws, but not certainly. You could miss it by bad luck. You can be fairly confident, not sure.
  4. Yes! You saw a blue block come out, so there is at least one in there. Seeing something once proves it is possible.
  5. So that the bag stays the same for every draw. If you keep blocks out, the chances change after each draw, and the bag soon runs out.
  6. Bag Q. Bag Q would give about 8 orange in 10 draws, and 7 is close to 8. Bag P would give about 2, and 7 is very far from that.
  7. Mia's. 100 draws wobble much less (compared with the total) than 10 draws. Mia's guess is about 7 orange in
  8. Max's 6 out of 10 does not disagree with that much; it is just less certain.

Common mistake

Being too sure after a few draws. Three oranges in a row do not prove the bag is all orange. Small experiments wobble a lot. Draw many times before you guess, and say how sure you are.

Forgetting to put the block back. Then each draw is from a different bag, and the tally no longer tells you the share in the bag you started with.

Where it leads