Question

Lottery statistics question.

In the New York State Lottery there is a game called Win 4. The object of the game is to select, in order, each of the correct four single digit numbers 0-9 on the random drawing (i.e. each of the 4 numbers will be drawn independently and will result in a 4 digit number from 0000 to 9999 which you must get the numbers in the correct order to win). Suppose you want to play the Win 4 game the next day but decide to seek out a fortune teller to give yourself a little "advantage" to the next day's drawing. The fortune teller, who is never wrong, says they will give you "part" of the information of the next day's drawing and gives you two choices of what information they will provide you... Choice 1: The fortune teller will give you the first two numbers (in the correct order) and will tell you whether the third and fourth numbers are even or odd (0 will be considered even for this purpose). So for example the fortune teller could say The first number is 3 The second number is 8 The third number is even The fourth number is odd Choice 2: The fortune teller will give you all four numbers but not necessarily in the correcet order. So for example the fortune teller could say the numbers are 3,8,5, and 1 where the actual winning number would be one arrangement of those numbers. Of the two above choices which one would result in the best chance of winning the next day's Win 4 game? You must support your answer with proper mathematical reasoning of why your choice is the correct one (i.e. show that it has the best probability of success).

Answer 1

Choice 1

We have 3 in first place, 8 in second place and 3rd and 4th place is empty.

3rd place can be 1 from any 5 even numbers.

The 4th place can be 1 from any 5 odd numbers.

$\\ \text{Prob of correct 4 digit number}\\ =\frac{1}{1*1*nCr(5,1)*nCr(5,1)}\\ =\frac{1}{1*1*5*5}=1/25$

Choice 2

We have 4 numbers. 3,8,5,1

But we don't know the order of the numbers. Only one order among all permutations is correct

$\\ \text{Prob of correct order of 4 digit number}\\ =\frac{1}{nPr(4,4)}\\ =\frac{1}{24}$

So choice 2 is a better option as it has more chance of correctness.