Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

Do You Feel Lucky?

Stage: 3 and 4 Challenge Level:

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.

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.

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!