# Settle an Argument

#### ScottWH

##### New Member
A social club has a daily contest where they draw a name from a list of all of the club members. A name is drawn each day. In order to win, you must visit the club and sign the book the day of the drawing.

There are 1600 club members. Let's say on Monday 300 members visit the club and sign the book. What are the odds of winning the Monday drawing? Are the odds less than or greater than 1 in 1600?

Thanks.

#### Dason

So we're interested in the Monday drawing... and you just told us that only 300 people signed up for the monday drawing. Right? Does the fact that there are 1600 club members even matter if only 300 people signed up?

#### ScottWH

##### New Member
The name is drawn from the list of 1600 and not from the list of 300. I'm not certain if the 1600 club members matter. I suspect they do.

#### Dason

Ok, but you said you need to have signed the book on monday to win. So even if they pick a name for somebody that didn't sign up on monday don't they just have to throw that away? Basically reducing the problem down to who signed the book on monday?

##### Ninja say what!?!
Even though the name is drawn from the list of 1600, the 1300 who did not sign the book are not eligible. Their names would just be tossed if drawn, and the club house would redraw. Because the probability of each person being drawn is the same, and the 1300 are thrown out, you have 300 eligible people with one drawing.

#### ScottWH

##### New Member
If a name is drawn of a person who did not sign the book, no one wins and it all starts over the next day.

#### ScottWH

##### New Member
So does that mean if 300 people signed the book on Monday, then there is a 1 in 300 chance of winning even though they are picking the name from the list of 1600 members?

#### Dason

I don't think we really have enough information on how this particular lottery is run. So let's get the facts straight.

You need to sign the book on the day they run the lottery to have a chance of winning.

1600 members
300 sign it on monday

They pick the winner from a list of all 1600 members. If the person they select didn't sign the book on monday then there is no winner that day.

What happens after that? On Tuesday do they run the lottery once again using the list of people that signed the book on Monday? Or do they run it again and only people who sign the book on Tuesday are eligible to win?

(If I got any of the facts wrong let me know).

#### ScottWH

##### New Member
You are correct. Only people who sign the book on Tuesday are eligible to win Tuesday's drawing.

There is a drawing once a day. When you sign the book, you must pay one dollar. The money goes into a kitty. If someone wins, they win the money that is in the pot. If no one wins on Monday, then the money rolls over to Tuesday and so on. So if someone wins on Tuesday, the money is paid out and the house contributes 100 dollars to the kitty and the process starts over again. You can only win if you sign the book the day of the drawing.

My contention is that since they draw the name from the full member list (1600 members) and you must sign the book the day of the drawing, then the odds of winning are less than 1 in 1600. Obviously, if there was not a book to sign and they pulled a name every day, the odds of winning are 1 in 1600. If they pulled the names from only the people who signed the book, then the odds would be 1 in how ever many people signed the book or for our example the odds would be 1 in 300.

So what are the odds of winning if they pull the name from the full member list and you must sign the book to win? Are they less than 1 in 1600?

Thanks.

#### squareandrare

##### New Member
The probability of winning, given that you've signed the book, is 1/1600.

#### Dason

Well. Given that a person signs the book on that day then their probability of winning is 1/1600. Given that a person doesn't sign the book on that day then their probability of winning is 0. If you want to the know probability of winning without knowing whether they signed the book on that day then we need to at least know the probability that that person signs the book on any given day.

#### ScottWH

##### New Member
I don't understand how the probability of winning is 1/1600 when you have to sign the book to win. Isn't there a probability of your name being drawn from the member list which is 1/1600 and then isn't there a second probability of your name being in the book?

#### Dason

... But you're the one who signs the book so you know whether you signed it or not.

What I was saying was that if we KNOW that you signed it on a given day then the probability that you win is 1/1600. If we KNOW that you didn't sign it then the probability of winning is 0. If we don't have that information and you're just asking "What's the probability that Tom will win today" and then I ask "Well did Tom sign the book?" and you respond "I don't know" then we need to know the probability that Tom will sign the book on any given day to answer your question.

Edit: But once we know the probability that they sign the book on any given day we can say without knowing whether that person signed the book that their probability of winning is P(Signed the book)*(1/1600). So if we assume that a person doesn't sign it every day (which would give P[signed the book] = 1 ) then the probability a given person will win the lottery that day (not knowing if they signed or not) is lower than 1/1600.

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#### ScottWH

##### New Member
Thanks for your answers and patience, but I just don't get it. Maybe I'm not asking the question correctly. I would like to know what the odds are of anyone winning the daily drawing.

Doesn't it matter how many people sign the book? If one person signs the book the odds of someone winning has to be less than say if 1000 people sign the book. Correct? If this is a true statement, then how can the odds of winning for those people who signed the book be 1/1600?

##### Ninja say what!?!
The explanations are pretty clear. Your concern of whether it matters how many people sign the book is addressed with the probability that a person signs the book. Like Dason said, if you don't know whether the person signed the book or not, the probability of winning is P(signed the book)*(1/1600).

Edit:
so if only 300 people signed the book on Monday, then the probability of someone winning is (300/1600)*(1/1600).

#### TheEcologist

##### R purist
Thanks for your answers and patience, but I just don't get it. Maybe I'm not asking the question correctly. I would like to know what the odds are of anyone winning the daily drawing.

Doesn't it matter how many people sign the book? If one person signs the book the odds of someone winning has to be less than say if 1000 people sign the book. Correct? If this is a true statement, then how can the odds of winning for those people who signed the book be 1/1600?
You are correct!

The calculation: 1/1600 x 300/1600 is not the probability of someone winning.

The calculation as Link supplied them is approaching the problem from the point of view of a random club member, 'playing against the all other members' ie what is the probability that any random club member gets his name drawn and has signed in. I think your previous wording confuses people as to what you are really after.

Is this then what you mean?

Calculating the probability of someone winning [that is anyone on a given Monday], is better explained from the point of view of the club; what is the chance that the club will give away a prize on a Monday - and yes, this is strongly dependent on the amount of 'playing members' ie the book signers.

The chance depends on the participants: If nobody signs, its 0 and if everyone signs its 1. The probability is also additive, each of the individual book signers on Monday has a 1/1600 chance of wining but 'the probability of someone winning' is the combined probability of them all: 300/1600)

It can then be calculated as: (1/1600) x N booksigners

When 300 people sign-in, its already high: 300/1600 (and of course when everyone signs its 1/1600 x 1600 = 1600/1600: they will have to give away a prize).

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#### ScottWH

##### New Member
So does an induvidual book signer have a 1/1600 chance of winning on Monday or is it (300/1600) * (1/1600)?

What is the difference in your two statments:

The calculation: 1/1600 x 300/1600 is not the probability of someone winning.
The calculation as Link supplied them is approaching the problem from the point of view of a random club member, 'playing against the all other members' ie what is the probability that any random club member gets his name drawn and has signed in.

The probability is also additive, each of the individual book signers on Monday has a 1/1600 chance of wining ....

##### Ninja say what!?!
I think there was a misunderstanding. Adding on to the Ecologist's comment, I hope to clarify this for you.

The probability of a specific person winning on Monday is:
$$P(signed \;the \;book)*P(being\; drawn)=(300/1600)*(1/1600)$$
That's for any random person and you do not know whether they signed the book or not.

The probability of the club giving away a prize is dependent only upon how many people out of the 1600 sign the book:
$$P(give \;out \; prize)=P(signed \;the \;book)=300/1600$$

The probability of someone winning a prize, given that you know they've signed the book is:
$$P(signed \;the \;book)*P(being\; drawn)=(1)*(1/1600)$$

Does that help?