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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?

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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?

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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: Challenge Level:1

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.