Do you feel lucky?
Problem
Do You Feel Lucky? printable sheet
Some people offer advice on how to win at games of chance, or how to influence probability in your favour for an easier, happier life. For each statement below, decide whether you think it is good advice, and use your mathematical understanding to justify your decisions.
Lottery advice: (for information on how lotteries work, click here)
Roughly equal numbers of odd and even are drawn most weeks, so you should pick a good mixture of odds and evens.
Choose six numbers with a total between 100 and 200, because the total is rarely outside this range.
Never choose six numbers all from the same group - for example, all single digits, all multiples of five, all with the same last digit...
Always pick some higher numbers from the 30s and 40s.
Coin Flipping:
If tails has come up on the last 9 occasions then it's a good idea to call tails again.
Winning at Roulette:
If red has come up lots of times in a row, you should bet on black next.
Snakes on a plane:
When you're flying, always take a pet snake with you in your hand luggage. The probability of there being TWO snakes on the plane is almost zero, so you will be safe from snake attack.
Staying dry at the cricket match:
Follow the example of the famous mathematician Hardy and take an umbrella with you to cricket matches. If you forget your umbrella it is more likely to rain, so if you remember to take it with you it is more likely to be sunny all day.
Send us your thoughts on these pieces of advice, as well as any other examples you can find of people giving unhelpful advice based on statistics and probability.
Getting Started
In the UK, lottery players choose six different numbers between 1 and 49. When the lottery draw takes place, 49 coloured balls numbered from 1 to 49 are placed in a machine and spun round, and then six balls are released at random. Any player who chose the six numbers picked by the machine wins the jackpot prize. Other lotteries use similar systems, perhaps with more or fewer numbers to choose from.
Can you think of examples where future events are influenced by past events?
Can you think of examples where future events are independent of past events?
Student Solutions
This is an open-ended question and we have received a great number of great responses from students.
Muntej from Wilson's School provided some real life observations for the lottery problems:
"Roughly equal numbers of odd and even are drawn most weeks, so you should pick a good mixture of odds and evens."
I have watched the lottery and it would rarely be a balance, the average over 1 year would be a balance but on a specific week the odds could be anything, and are usually biased to one side or the other.
"Always pick some higher numbers from the 30s and 40s."
Definitely not. On lots of the weeks, the numbers range between 1-30, picking between 30 and 40 limits chances of winning even further.
Elliot from Wilson's School submitted an excellent explanation on the lottery problems
Each sequence of six numbers is just as likely as the next, whether it is 1, 2, 3, 4, 5 and 6 or 5, 16, 22, 31, 37 and 49. This is because each number has a separate 1 in 49 (or 48 down to 44, as two numbers cannot be chosen twice) chance of being picked. It is true that sequence with mixed evens and odds are more likely than one with only odds or evens, because it has a larger pool of possible numbers to choose from. But each particular sequence will have equal chances of winning.
This is also true for the advice that you should never pick numbers from the same group. It is just as likely to be 5, 15, 20, 25, 30 and 35 as it is to be 2, 17, 26, 30, 34 and 48. Picking at least one number from 30 - 40 does not help either, as it is just as likely to be any number, such as 19, as to be 37. It is also false that you should pick numbers totalling 100 - 200, for similar reasons. Picking 3, 19, 33, 37, 45, and 46 is just as likely as 1, 3, 5, 7, 9, and 11 for example. So some lottery advice is true, but not useful.
Charlie and Nathan, also from Wilson school, pointed out that the result of the next coin flip or Roulette colour does not depend on the previous ones:
The coin advice is also wrong as it could land on heads 99 times but the next time the chances of heads and tails will not be affected. For a fair coin, this probability is 50-50. On Roulette the numbers and colours are random so even if it has been red nine times it could be red again because it is pure chance.
Emerson and Chris from St Peter's School summarised the point by saying
It doesn't matter what the last time was you still have a fifty fifty chance in the next try. So the coin flipping statement is wrong as it is always has an equal chance of being heads or tails. The roulette is wrong as it is always has equal chance of being red and black.
Conner from Gladesmore and Janusz from Wilson's also submitted correct comments.
Does bringing another snake onboard the plane reduce the chance of a snake attack? Or does bringing an umbrella to a cricket game reduce the chance of rain?Charlie from Wilson's School said:
Bringing a snake with you onto a plane would not decrease the chances of another snake coming onto the plane as well. And the chances of it raining are exactly the same if you bring or do not bring an umbrella.
Eliotte from Wilson's School added:
Bringing an umbrella to cricket games does not affect the weather in any way, so it is just as likely to be sunny or rainy if you bring your umbrella or not.
Adam from Totton College gave a very clear and comprehensive reasoning to the question above:
"Roughly equal numbers of odd and even are drawn most weeks, so you should pick a good mixture of odds and evens."
On average, there will be one more odd ball in every 49 balls picked compared with even. Therefore, it would be better to be slightly biased towards odd numbers, but to have a good mix of odds and evens as well.
Choose six numbers with a total between 100 and 200, because the total is rarely outside this range.
The true range of totals is between 21 and 279 , making the average tend towards 150. Going 50 either way would catch most possibilities, therefore this is good advice.
Never choose six numbers all from the same group - for example, all single digits, all multiples of five, all with the same last digit...
Impossible. Every number between 1 and 49 belong in the same group, therefore this can?t be done. Always pick some higher numbers from the 30s and 40s.
It doesn't really matter what balls you pick, as you'll have the same chance no matter what you do, but making sure that you cover a lot of ground can sometimes give you a chance.
If tails has come up on the last 9 occasions then it's a good idea to call tails again.
I would call Heads on this occasion. Besides, there's truly a 50-50 chance of getting Heads compared to Tails.
If red has come up lots of times in a row, you should bet on black next.
Again, there is a 50-50 chance (minus a bit due to the 0 and 00) that black will come up. Personally, I would agree with this if it's to average out, there needs to be some blacks.
When you're flying, always take a pet snake with you in your hand luggage. The probability of there being TWO snakes on the plane is almost zero, so you will be safe from snake attack.
Bad advice. The more people who follow this, the more likely there will be two or more snakes on the plane.
Follow the example of the famous mathematician Hardy and take an umbrella with you to cricket matches. If you forget your umbrella it is more likely to rain, so if you remember to take it with you it is more likely to be sunny all day.
Coincidences make you think that this is true. In fact, it doesn't matter if you bring an umbrella or not it has no effect on what the weather does.
Well done Adam, thank you for providing such a complete response.
We have received a lot of good answers for this question. Conner from Gladesmore, Charlie, Nathan, Muntej, Janusz and Ayobami from Wilson's all pointed out the independence of rain from whether we take an umbrella or not. This is one key idea in the study of probability. Well done to you all!
Teachers' Resources
Why do this problem?
This problem is one of a set of problems about probability and uncertainty. Intuition can often let us down when we meet probability in real life contexts; this problem has been designed to provoke discussions that challenge commonly held misconceptions such as the Gambler's Fallacy.
Possible approach
Hand out this worksheet (Word, PDF) with the statements from the problem, and give everyone time to read them through and decide for themselves whether they think the advice is good or not.
Key questions
Are future results affected by previous results?
Lots of advice is based on accurate statistical data - does that necessarily mean it is useful advice?
Possible support
The problem has been structured as a discussion task so that learners can support each other in coming to a better understanding. By allocating a view for each pair to argue, it allows those who hold these misconceptions the chance to freely explore them without fear of ridicule.
Possible extension
Ask learners to collect over several weeks some examples of probability misconceptions in the media, in school or at home, which could be used to create a classroom display.