Bingo Cards

BrianCalgary

I have always been interested in probability. But I missed the course in High School where it was taught. I was volunteering at a Bingo hall on Saturday and got to wondering how many different Bingo cards there are.

There are 5 columns. 4 columns have 5 squares and 1 column has 4 squares with the 5th square being in a fixed position. Each column has a choice of 15 different numbers that can go into any square, but the number can only appear once.

Question 2 has to do with general probability, is there some sort of formula to figure it out. As an example, how many differnt combonations in a lottery with 49 different numbers and 6 chosen? how do we figure out th eodds against winning?

Thanks for the help.

Tar.Aldarion

2)

There are 49 numbers, the first number comng out as you want is 1/49.
The 2nd will be 1/48, then 1/47, 1/46, 1/45, 1/44 as the amount of numbers left will be decreasing.
You assume what you wanted to happen, did happen.
You multiply these six numbers together(as you want them all to happen for you to win) , giving you a very unlikely chance of winning the lotto.
If you want something AND something AND something AND something to happen, you multiply the probabilities together.
With different problems there will be different methods to doing it, probablity q's are all logic really.

To get the odds against winning you minus the tiny number you get, from 1(think of 1 as 100%).
1 means something happening, 0 means it does not. You will get .999999....
chance against winning etc.

I am in the middle of a poker tournament and will get back to you on the first, I wrote this fast, hope it makes sense.

May aswel edit here too: Where I said 1/49 it should be 6/49 as the order does not matter in lotto, unlesl you are in some dodgy one. It follows on that it will be 5/48, 4/47, 3/46, 2/45, 1/44.

rjt

Well, there isn't any singular formula that will answer all probability questions, but there are a couple of key operations that will answer most.

The first two basic rules with probability are that AND is the same as multiplying and OR is the same as adding. eg. the if the probability of it raining is 1/5 and the probability of it being windy is 2/5, then the probability of it being wet AND windy is (1/5)x(2/5) = (2/25). And the probability of it being wet OR windy is 1/5 + 2/5.

If some event has x possible outcomes, and we want to know the probability that the outcome will be one of y different possibilities, the probability is y/x. Eg. if we roll a dice, there are 6 possible outcomes. If we want to know the probability of it being a six, we know the answer is 1/6. If we want to know the probability of the dice showing a five or a six, then we have 2/6 (as y is now 2, as there are two outcomes we are looking for, a five or a six).

Now, a useful function is the nCr function (where n and r are some numbers, with n greater than or equal to r and neither being negative). nCr is the number of ways to choose r unique objects from a group of n. Eg. if there's a group of 15 football players, the number of different 11 man teams I can pick is 15C11. Hence, in the lotto, the number of ways you can choose 6 from 49 is 49C6, which a calculator shows to be 13983816. This is the number of different choices you have when you pick numbers. Now, one of these will be the winning combination. Hence, (as mentioned in the paragraph above) the probability of winning the lotto is 1/13983816.

(Also, if you're interested, to calculate nCr by hand, use the formula
(n!)/(n-r)!(r)!. where n! = (n)x(n-1)x(n-2)x...x(1)
So 3C2 = 3!/(1!)(2)! = 6/(1x2) = 3
This is straightforward enough to prove.)

As for the first problem, I'm not quite sure I understand you completely, but from the way I read your post: in the first column, we have to choose 5 numbers from 15 (I'm assuming that order doesn't matter, so that if the first column is 1,2,3,4,5, this is the same as 5,4,3,2,1). This is 15C5. Same with the next two columns. The last column has only 4 spaces, so we have 15C4 possibilities. There's a rule that says if you have to pick one object out of a total of A objects, and then pick another object out of a total of B objects, the number of different ways you can take objects is (A x Y). The same thing applies here, as we have to pick numbers for the first column, followed by numbers for the second column, then the third and then the fourth. So we have a total of (15C5)x(15C5)x(15C5)x(15C4) = 3.7x10^13 different bingo cards.

Tar.Aldarion

Ah, just won the poker, woo money.
Sorry, did not notice the combinations bit, the formula as said above is nCr.

rjt, is 13983816 not just the amount of combinations, whereas the probabilty is different(1/1006834720)?

You find this using the formula, n! / (n - r)! r!, or more simply, as I said in my first post.

Therefore the chance of you not winning Brian is 1 - 1/1006834720.
Do you really want to know how slim the chances are?
Stick to bingo to win big. (:

Ah, I'm so tired, I forgot to mention(assuming you have picked 6 numbers), since order does not matter, we can divide this number by how many ways these numbers can be arranged. There are six possibilities for the first ball, five for the second, 4 for the third, 3, 2, and one left over. That is 6 × 5 × 4 × 3 × 2 × 1 = 720 So, divide the odds I mention above by 720, which makes the above post by rjt right.

rjt

Tar.Aldarion wrote:

Ah, just won the poker, woo money.
Sorry, did not notice the combinations bit, the formula as said above is nCr.

rjt, is 13983816 not just the amount of combinations, whereas the probabilty is different(1/1006834720)?

You find this using the formula, n! / (n - r)! r!, or more simply, as I said in my first post.

I don't think we use nPr, as that counts 1,2,3,4,5,6 as different numbers to 6,5,4,3,2,1. (and both are equal in the lotto). The problem with what you said in your post is that the probability of getting the first number correct is 1/49, whereas it should be 6/49 (as the first number will be part of the winning lotto numbers if it is the same as any one of the six winning numbers).

For example, with your method the probability of winning in the Irish lotto is 1/3776965920, yet nearly every time someone wins (and at most 10,000,000 tickets are bought).

Tar.Aldarion

Yep, sorry, noticed a minute ago and edited!

EDIT: On to the bingo thing, I don't know bingo well and I am not sure what you are asking at the moment.

25 squares on a Bingo card, 5 x 5 square. The centre space is free which leaves 24 spaces that may or may not have been matched in the course of a game. This gives 2^24 = 16777216 patterns that may or may not have at least 1 Bingo. Each of these is an equally likely pattern. For each of these patterns, we are going to have to count how many spaces are occupied and if the pattern contains a bingo. EG: There are 24C11 = 2496144 different ways we can hit 11 different spaces on the Bingo Board. We will have to count how many of these 2496144 have at least one Bingo. Then we can divide this count by 2496144 which will give us the probability of having a Bingo.

What do you want to know, the probablility of getting a bingo after a certain amount of turns/hits??

BrianCalgary

Tar.Aldarion wrote:

What do you want to know, the probablility of getting a bingo after a certain amount of turns/hits??

No, just the number of possible cards. There are so many different ways to win depending on the game that to ask that question without nailing down this first point would break my brain.

Thank for your responses. Now to take pencil to paper and run some tests myself and see how I did.

Congrats on winning the poker tourney. I'll play with a math whiz.:)

Tar.Aldarion

Thanks.
So, if I understand, rjt answered your question, and all is well!
Feel free to enquire about more maths ponderings.

rjt

Tar.Aldarion wrote:

Thanks.
So, if I understand, rjt answered your question, and all is well!
Feel free to enquire about more maths ponderings.

I don't think I did. My answer assumed that every box has a number in it, but thinking about that, I don't think that's true. I can't remember bingo at all

Tar.Aldarion

Don't 24 boxes have a number in them at the start?
(I see we are not up to date on the rules of bingo)

As far as I understand...
Column
1, 1-15
2, 16-30
3, 31-45
4, 46-60
5, 61-75
With the central square free.
There are (15!/10!)^4 * (15!/11!) = 5.52446474 × 10^26 (over 552 million billion billion) possible combinations that could exist.

Edit: Thinking about it, I think I am right, but I always think that. (:

BrianCalgary

Tar.Aldarion wrote:

Don't 24 boxes have a number in them at the start?
(I see we are not up to date on the rules of bingo)

As far as I understand...
Column
1, 1-15
2, 16-30
3, 31-45
4, 46-60
5, 61-75
With the central square free.
There are (15!/10!)^4 * (15!/11!) = 5.52446474 × 10^26 (over 552 million billion billion) possible combinations that could exist.

Edit: Thinking about it, I think I am right, but I always think that. (:

Now I'm better rested. My daughter was called up for a football tourney, first game this am, they won 2-1, she played well, my wife said she loved me, all is right with the world.

Your description of the card is correct.

I don't understand what 15! means. I guess I need to know what that represents in a long version.



As an aside: when I visit Eire where should I stay. I was at a lovely B&B in Blarney and another in Rosslare.

rjt

BrianCalgary wrote:

Now I'm better rested. My daughter was called up for a football tourney, first game this am, they won 2-1, she played well, my wife said she loved me, all is right with the world.

Your description of the card is correct.

I don't understand what 15! means. I guess I need to know what that represents in a long version.

15! = 15x14x13x12x11x10x9x8x7x6x5x4x3x2x1 = 1307674368000

And good to hear that all is right in the world

Tar.Aldarion

BrianCalgary wrote:

Now I'm better rested. My daughter was called up for a football tourney, first game this am, they won 2-1, she played well, my wife said she loved me, all is right with the world.

Your description of the card is correct.

I don't understand what 15! means. I guess I need to know what that represents in a long version.



As an aside: when I visit Eire where should I stay. I was at a lovely B&B in Blarney and another in Rosslare.

As said above, it is just multiplying all the positive numbers decrementing to one.
http://en.wikipedia.org/wiki/Factorial
Ah, good for your daughter, and I suppose it is good that your wife loves you.



Well, where do you want to stay in Ireland and how fancy do you want it to be?