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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!