Calculating total possible matching combinations remaining

[charles1234]

Hello

I'm working on an application and I have a problem that I've been wrecking my brain over. The best way I can explain it is through a game.

There are 4 bags with numbered balls in each (the total number of balls in each bag is irrelevant). You have to guess the number that is going to be selected from each bag and you will win a prize if you match all 4 selections and also if you match 3 out of 4. You are also allowed to play combinations, meaning you can have multiple guesses for one or more of the bags. So for example, you can play a ticket that looks something like this:

Bag one: 1, 5
Bag two: 4, 6
Bag three: 1, 9
Bag four: 7, 9

In this case the ticket has 16 different combinations (2x2x2x2). I've used two selections in each for simplicity but there could be an arbitrary amount in each.

The question is: How can I calculate the remaining possible winning combinations in each prize category after a ball has been selected from the 1st bag, then 2nd bag etc.

So for example:

First ball chosen is number 1. The answer would be 16 combinations for 3 out of 4 and 8 combinations for 4 out of 4.
Second ball chosen is number 3. The answer would be 8 combinations for 3 out of 4 and 0 combinations for 4 out of 4.

The only way I've managed to do this is by breaking it up into all possible combinations and then calculating the possible matches on each one but is there an easier way without having to analyse every combination?

Thanks

 

[jvicens]

Hello,

Thank you for sharing your problem with us. It seems like you are trying to find a more efficient way to calculate the remaining possible winning combinations in each prize category after a ball has been selected from each bag. I would suggest looking into the concept of probability and using mathematical equations to solve your problem.

First, let's define some terms to make it easier to understand. The total number of balls in each bag is called the "sample space". The number of balls that match your guess is called the "successes". And the number of balls that do not match your guess is called the "failures".

To calculate the remaining possible winning combinations for 3 out of 4, you would use the formula: (number of successes)^(number of bags - 1) x (number of failures)^(number of bags - number of successes). In your example, the first ball chosen is number 1, so the number of successes is 2 (one in bag one and one in bag three) and the number of failures is 6 (two in bag two and four in bag four). Plugging these numbers into the formula, we get (2)^(4-1) x (6)^(4-2) = 8 x 36 = 288 possible combinations for 3 out of 4.

For 4 out of 4, the formula would be: (number of successes)^(number of bags). In your example, there are no successes for the second ball chosen, so the number of successes is 0. Plugging this into the formula, we get (0)^(4) = 0 possible combinations for 4 out of 4.

I hope this helps you with your problem. Let me know if you have any further questions or need clarification. Good luck with your application!