### 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?

### In a Box

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

# Scratch Cards

##### Stage: 4 Challenge Level:

Here is another nut that has been waiting to be cracked for months. Three solutions arrived simultaneously from Bradley, Avery Coonley School, USA; Joe, Anglo Chinese School, Singapore and Ling Xiang Ning, Allan, Tao Nan School, Singapore, whose solution is given below.

Mail order companies often send a scratch card with their catalogue. These cards are used to encourage people to order from the catalogue by offering a free price. You have to uncover three of the numbers by scratching off the cover and revealing the number under three of the circles. Of course the numbers under the circles are randomly placed. If the numbers add up to more than fifteen you win a prize. What is the probability of winning?

As there are 9*8*7 possible choices of 3 numbers, and the order of revealing the numbers does not matter, the number of choices of the three numbers is (9*8*7)/6 = 84

The probability of winning is the number of winning combinations divided by 84.

We have to count the number of winning combinations:

We start from 987, 986? and the first number chosen gets smaller, and the other numbers must be smaller than the first so that three numbers will not appear together more than once.

There are total 38 choices, as shown:

987, 986, 985, 984, 983, 982, 981, 976, 975, 974, 973, 972, 971, 965, 964, 963, 962, 961, 954,
953, 952, 943, 876, 875, 874, 873, 872, 871, 865, 864, 863, 862, 854, 853, 765, 764, 763, 754

So, the probability of winning a price is 38/84 or also, 19/42.