Misunderstanding Randomness
Which of these ideas about randomness are actually correct?
Age
11 to 14
Challenge level
Problem
Which of these statements do you agree with?
a) I've renumbered the faces of an ordinary die to read: 2, 2, 2, 2, 2, 9. I roll it a lot and write down the numbers; they are random numbers.
b) I flipped a coin 10 times and got all heads, something must be wrong with the coin or my throwing.
c) I've flipped a coin 10 times so far and got heads every time. It is really likely that the next one will be a tail because the tails really need to catch up.
d) I've flipped a coin 10 times so far and got heads every time. I know the next one will be heads because the coin must be biased.
e) I did 3 divided by 17 on my calculator, I got 0.1764705882... I think the decimal digits are random.
f) I pressed pi (symbol) on my calculator and got 3.141592654... My teacher told me that the decimal goes on forever without any pattern, this means the digits are random .
g) I want to know how tall I'll be when I grow up, but no-one can work it out in advance; we just have to wait and see. This means it's random.
h) The weather forecaster said there is a 30% chance of rain in my region tomorrow
- So it will rain on 30% of the region.
- It means that it'll rain for 30% of the time.
- Out of all the regions which get this warning, it'll only rain on 30% of them.
- 30% of the forecasters think it'll rain, the other 70% think it'll be sunny.
i) The weather is totally random - it's so mixed, and even if the forecasters try very hard, they don't always get it right.
j) The weather tomorrow isn't random, it depends on the weather today.
k) In the national lottery it is very likely that the balls selected will be a mixture of colours. You are more likely to win if you select numbers from a mixture of colour sets.
l) In the national lottery, you shouldn't pick a set of consecutive numbers because it's almost impossible for them to come up.
m) Ann: Here's a truly random sequence (thinking "I got them from throwing a die"...) 1, 5, 3, 4, 2, 6, 6, 3, 3, 5,...
Bob: But they're not random - there are no sevens or eights or nines!
Chris: You're both correct! Randomness is all about perception - what matters is how you view the numbers, not how they were generated.
Getting Started
Some definitions
random:
A term used in probability, relating to things that happen by chance. It means that there is no special pattern to the items or numbers that are selected or chosen, that is, they happen in a haphazard way.
random experiment:
Some process that is carried out according to a set of rules. The rules determine completely the process that we carry out, but they do not tell us in advance what the outcome of the experiment will be. A random experiment can be repeated arbitrarily often.
random number:
A sequence of randomly generated numbers, where all the numbers had an equal chance of being picked, every time. Tables of random numbers used to be printed in books, these days they're usually simulated on computers/calculators. They are used to select random samples in statistics.
Student Solutions
Patrick from Woodbridge School sent us the following comments:
My definition of "random" that I use here is "involving equal chances for each number".
In fact, mathematicians use the word 'random' to mean something happens, or is chosen, by chance, and not by a rule or a pattern.
When does this agree/disagree with Patrick's comments?
I believe that:
a) This is not random, as clearly the number 2 has a far higher chance of coming up.
b) This is not a mistake - the chances are 1/(2^10) that this could happen, so it is unlikely but perfectly possible
c) This is not true - there is still a 1/2 chance of this happening, the coin is not conscious.
d) This is possible but not true - again, there is a chance (albeit quite small) that this could happen.
e) This is possible but we only have a small sample of the number so it could repeat after 16 decimals, for example.
f) This is true - if there is no pattern to the digits.
g) This is very nearly but not quite random - it is influenced by some environmental factors (an extreme example is growing up with a heavy weight on your head!).
h) i) This is not necessarily true - the rain might still happen but might cover the whole region.
ii) This is not true - there might be huge downpours all the time, that were unexpected, to generate the 30% chance.
iii) This is true assuming the weather behaves, and the weather forecaster has given out the same warning.
iv) This is a slightly odd way of making the statistic, but it could be used.
i) The weather is not totally random - for example, a drop in air pressure is linked to thunderstorms.
j) This is partially true - some remnants of the weather will affect the weather for tomorrow.
k) This is wrong - the balls are totally random. There is the same chance of winning withballs of one colour as winning with balls of varied colours.
l) This is wrong - it is almost impossible to choose any correct sequence. There is no more likelihood of the sequence being 1, 2, 3, 4, 5, 6 than 11, 24, 28, 34, 41, 45.
m) Chris is right - Anna is right using my definition, but Bob is right using a definition of "allowing equal chances of all digits".
Teachers' Resources
Why do this problem?
All work on probability is based on ideas of randomness, an idea which has precise mathematical meaning, while being informally used in everyday life. A discussion of tricky ideas should challenge students' understanding.
Possible approach
Stand rolling a die, or shuffling some cards as the students enter, or while the discussion starts. If you have an interactive whiteboard, you could leave a slideshow running the numbers1-10 set to shuffle and loop. Ask students for initial ideas of what randomness means, and a show of hands for how many already understand it.
Put students into small groups (randomly?) and ask them to compose one sentence that explains randomness. Organise a 'random relay' - give each group a slip of paper with one of the statements from the problem on it. They must decide whether they agree with it or not, and settle on the main arguments in case they are called on to argue with a group with the opposite opinion. They then get a new statement to work on. Make it very clear that the whole group is responsible for the answers, and that any of them might get called to explain. Tell them that the points available are 10 for every right answer, -20 for each wrong one, (to promote certainty above speed).
Select a few items for debate where groups have reached different final answers. Either arrange for representatives of the groups to meet and convince each other, or arrange a public debate with one champion from either view, and then questions from the floor.
The final scores for each group might reflect the general misunderstanding of probability in the general population, the ideas are difficult, but we can make sure that we understand the basics - reiterate them from the board.
Key questions
How does your justification relate to the original statements made about randomness? (perhaps worth displaying these on the board throughout the lesson)
Possible extension
Focus on clarity of explanation and attempt to get good answers on all statements.
Possible support
Students could start by working with interactivities like
Interactive
Spinners and Flippin'
Discs in order to establish the following key ideas:
- If something is random, you can't ever work out what the next one will be
- Even if it has been done lots of times, the next one could still be anything
- When it's been done lots of times the overall results are very predictable