I have started reaching out to math professors for my answer. Here is the first response: This was more compicated than I first thought.
Esteemed Math Aficionado,
Can you solve this mathematics problem?
The subject is a raffle; the problem is to determine the odds of winning at the onset of the raffle and during the raffle as variables change.
Facts:
• Only 300 tickets will be sold.
• Each ticket has 12 chances to win. (The raffle is yearlong, and a prize is awarded every month for 12 months).
• The cost of a ticket does not change (it will be the same cost no matter when a person enters the raffle).
• All 300 tickets (winners included), are returned to the drawing for all 12 drawings. (It is possible, though highly improbable, for 1 ticket to win ALL 12 prizes (Unless a ticket is bought later in the raffle)).
The answers needed:
Q1. What are the odds of winning 1 prize with 1 ticket?
First, "odds of winning" and "probability of winning" are different things. It's easier to work with probabilities. The probability of winning on any draw is 1/n, where n is the number of tickets in the draw. (Odds of winning would be "1 to (n-1)". )
The probability of winning depends on the number of tickets that are "in the drawing", i.e., in the container from which the tickets are taken when the winners are chosen. I gather that not all tickets are expected to be sold before the first drawing and that unsold tickets will not be in the drawing. Imagine the (unlikely) scenario where only one ticket has been sold before the first drawing. Then the holder of that ticket will certainly win. If this ticket remained the only one purchased for several draws, then it would certainly win all of them.
The other possibility is that you plan to put even the unsold tickets into the draw, (and award no prize if an unsold ticket is drawn?). In this case, each ticket has a 1/300 chance of winning on each draw.
I will assume that only the sold tickets go into the draw. On ecah draw, the chance of any particular ticket winning is 1/N(i), where
N(i) is the number of tickets sold before the i-th draw.
Q2. What are the odds of winning 2 prizes with 1 ticket? And subsequently, 3 with 1; 4 with 1; and so on?
This is difficult to calculate without knowing N(i).
However, if all tickets are sold before the first draw (i.e., N(i) =
300 for all i), there is a formula. Suppose X= number of wins. Then the probability that X=12 is (1/300)^(12), (a very small number). The
probability that X=0 is (299/300)^(12), which is approx. 96/100. So,
there's about a 4/100 chance of winning at least once.
There is a formula for probability(X=j). It's called the "binomial formula" or "binomial distribution". You can look it up on the web.
Using this, we get the following table (assuming ALL tickets are sold before first draw):
j, P(X=j)
0, 0.960725
1, 0.0385575
2, 0.000709252
3, 7.90694*10^-6
4, 5.95004*10^-8
5, 3.18397*10^-10
6, 1.24235*10^-12
7, 3.56144*10^-15
8, 7.44449*10^-18
9, 1.10658*10^-20
10, 1.11028*10^-23
11, 6.75146*10^-27
12, 1.88168*10^-30
Q3. How are the odds affected if a ticket is bought later in the year (as in 6 months later), after 6 prizes are already gone? Is there an equation that shows the decrease in odds as various numbers of tickets are purchased later at differing times of the yearlong raffle.
Yes. If you knew the function N(i), I could compute the probability of a win for any ticket.
Q4. Is there an equation or a number that shows what each decrease in odds is per ticket if some tickets are not sold out in advance of the first drawing?
Yes. But this is dependent on N(i). If you imagine that there will only be a very small number of unsold tickets, we could estimate.
Clarification of what I am trying to ascertain:
I want to be able to state what the odds are to prospective raffle participants before and during the contests. It is my goal to be able to announce the odds as the variables of the contest changes. Obviously, the odds of winning a prize are the best if a person participates at the onset BEFORE the first drawing, but know that the odds must worsen as the number of prizes diminishes. Surely, a raffle participant that comes on late in the raffle (say 6 months later), still has a decent chance of winning a prize since only 300 tickets are in the drawing, though that chance has to have been diminished since there were 6 less prizes available to that ticket as opposed to others who had 12 items present; or 6 additional chances.
Esteemed Math Aficionado,
Can you solve this mathematics problem?
The subject is a raffle; the problem is to determine the odds of winning at the onset of the raffle and during the raffle as variables change.
Facts:
• Only 300 tickets will be sold.
• Each ticket has 12 chances to win. (The raffle is yearlong, and a prize is awarded every month for 12 months).
• The cost of a ticket does not change (it will be the same cost no matter when a person enters the raffle).
• All 300 tickets (winners included), are returned to the drawing for all 12 drawings. (It is possible, though highly improbable, for 1 ticket to win ALL 12 prizes (Unless a ticket is bought later in the raffle)).
The answers needed:
Q1. What are the odds of winning 1 prize with 1 ticket?
First, "odds of winning" and "probability of winning" are different things. It's easier to work with probabilities. The probability of winning on any draw is 1/n, where n is the number of tickets in the draw. (Odds of winning would be "1 to (n-1)". )
The probability of winning depends on the number of tickets that are "in the drawing", i.e., in the container from which the tickets are taken when the winners are chosen. I gather that not all tickets are expected to be sold before the first drawing and that unsold tickets will not be in the drawing. Imagine the (unlikely) scenario where only one ticket has been sold before the first drawing. Then the holder of that ticket will certainly win. If this ticket remained the only one purchased for several draws, then it would certainly win all of them.
The other possibility is that you plan to put even the unsold tickets into the draw, (and award no prize if an unsold ticket is drawn?). In this case, each ticket has a 1/300 chance of winning on each draw.
I will assume that only the sold tickets go into the draw. On ecah draw, the chance of any particular ticket winning is 1/N(i), where
N(i) is the number of tickets sold before the i-th draw.
Q2. What are the odds of winning 2 prizes with 1 ticket? And subsequently, 3 with 1; 4 with 1; and so on?
This is difficult to calculate without knowing N(i).
However, if all tickets are sold before the first draw (i.e., N(i) =
300 for all i), there is a formula. Suppose X= number of wins. Then the probability that X=12 is (1/300)^(12), (a very small number). The
probability that X=0 is (299/300)^(12), which is approx. 96/100. So,
there's about a 4/100 chance of winning at least once.
There is a formula for probability(X=j). It's called the "binomial formula" or "binomial distribution". You can look it up on the web.
Using this, we get the following table (assuming ALL tickets are sold before first draw):
j, P(X=j)
0, 0.960725
1, 0.0385575
2, 0.000709252
3, 7.90694*10^-6
4, 5.95004*10^-8
5, 3.18397*10^-10
6, 1.24235*10^-12
7, 3.56144*10^-15
8, 7.44449*10^-18
9, 1.10658*10^-20
10, 1.11028*10^-23
11, 6.75146*10^-27
12, 1.88168*10^-30
Q3. How are the odds affected if a ticket is bought later in the year (as in 6 months later), after 6 prizes are already gone? Is there an equation that shows the decrease in odds as various numbers of tickets are purchased later at differing times of the yearlong raffle.
Yes. If you knew the function N(i), I could compute the probability of a win for any ticket.
Q4. Is there an equation or a number that shows what each decrease in odds is per ticket if some tickets are not sold out in advance of the first drawing?
Yes. But this is dependent on N(i). If you imagine that there will only be a very small number of unsold tickets, we could estimate.
Clarification of what I am trying to ascertain:
I want to be able to state what the odds are to prospective raffle participants before and during the contests. It is my goal to be able to announce the odds as the variables of the contest changes. Obviously, the odds of winning a prize are the best if a person participates at the onset BEFORE the first drawing, but know that the odds must worsen as the number of prizes diminishes. Surely, a raffle participant that comes on late in the raffle (say 6 months later), still has a decent chance of winning a prize since only 300 tickets are in the drawing, though that chance has to have been diminished since there were 6 less prizes available to that ticket as opposed to others who had 12 items present; or 6 additional chances.
