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GiveFive

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Jan 5, 2016, 4:00 pm

Hopefully someone smarter than I can answer a question regarding the odds for the Pennsylvania Millionaire Raffle and help to settle an argument at the same time.

A little background. Yesterday at work, the members of the pool that I run for the PA Millionaire Raffle were discussing the latest raffle. Someone said "The odds are 125,000 to 1" to which another person replied "No they're not. Not really" And that's when the argument got started.

Assuming all 500,000 tickets are sold, why are the odds not 500,000 to 1 when the first winning number of one million dollars is drawn, and after that, why are they not 499,999 to 1 for the second number?

After all, after the first winning number is drawn, it's excluded from the pool of 500,000 ticket numbers, and then there are 499,999 ticket numbers remaining that could be drawn. So why aren't the odds at that point 499,999 to 1? And after the second winning number is drawn, why aren't they now 499,998 to 1 for the third ticket?

The PA Lottery's website clearly states the odds are 125,000 to 1. But is that really true?? G5

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Murgatroyd

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Jan 5, 2016, 8:07 pm

There are at most 500,000 tickets, of which exactly four will win the top prize. If all the tickets are sold, the chances of a given ticket winning are exactly 1 in 125,000. If the deadline passes before all the tickets are sold, the chances of winning are better at 4 in however many tickets were actually sold.

You're right that if the top prizes are the first ones drawn, the chances of your ticket being the first, second, third, or fourth are 1/500000, 1/499999, 1/499998, and 1/499997 respectively. But the order doesn't actually matter, and when you do the calculations, the chances of getting one of the four do come to exactly 1/125000.

(Mathematical details: The probability of NOT getting the first one is 499999/500000; not getting the second is 499998/499999; not getting the third is 499997/499998; not getting the fourth is 499996/499997. To get the probability of not getting the first and not getting the second and not getting the third and not getting the fourth, multiply all those together - the result is 499996/500000. Thus, the probability that you will get one of them is 4/500000, or 1/125000.)
GiveFive

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Jan 5, 2016, 9:40 pm

Quote: Originally posted by Murgatroyd on Jan 5, 2016
There are at most 500,000 tickets, of which exactly four will win the top prize. If all the tickets are sold, the chances of a given ticket winning are exactly 1 in 125,000. If the deadline passes before all the tickets are sold, the chances of winning are better at 4 in however many tickets were actually sold.

You're right that if the top prizes are the first ones drawn, the chances of your ticket being the first, second, third, or fourth are 1/500000, 1/499999, 1/499998, and 1/499997 respectively. But the order doesn't actually matter, and when you do the calculations, the chances of getting one of the four do come to exactly 1/125000.

(Mathematical details: The probability of NOT getting the first one is 499999/500000; not getting the second is 499998/499999; not getting the third is 499997/499998; not getting the fourth is 499996/499997. To get the probability of not getting the first and not getting the second and not getting the third and not getting the fourth, multiply all those together - the result is 499996/500000. Thus, the probability that you will get one of them is 4/500000, or 1/125000.)

Thank you. Your explanation made perfect sense to me.

That said, I'm definitely not going to try to explain to the guys in my pool why it's so. See, we're a bunch of cranky old curmudgeons, and we'll throw down in a heart beat. Matter of fact I though that's exactly what was going to happen today when the argument began!!

I can see the headline in the local newspaper now; Old Fools Brawl at IBM Plant! Lotto argument causes punches to fly!

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dr65

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Jan 6, 2016, 6:30 am

Overall odds are that there are 4 winners in 500,000 tickets so simple math would calculate to 1/125,000

If the 4 top prize tickets were sold in the first 4 tickets, that would seem to throw the odds out of whack and that's

probably where the disagreement comes in at work.....they/you are thinking in terms of top prize winnings being sold right

off the bat..then what happens to the odds?...it makes some sense thinking like you do and if that scenario happened,

it would seem the odds are nowhere near the 1/125,000, that in fact, they must be MUCH better.

BUT...it's calculated among the 500,000 run and if you work in 4 winning top prize tickets...it translates to 1/125,000.

I could go hit an IVM and win a Million dollars....SO then, what odds was I playing with by putting in a $20 bill and

hitting one top prize out of 6 or 7 million tickets??? Seems like 1/1 but I got lucky that the top $1 million prize was

at that machine at the time I purchased it, still I'm playing a game where the odds are 4 top winners in a run of millions

of tickets. So I'm still playing at the posted odds...it's just that my purchase right then and there won me a million dollars.

That's probably what the argument is about...and it might never end lol

If some math wizard wants to come and explain what it seems like the odds would be if 4 people were to win the top

prizes in the first 2 weeks and say the lottery only released 1/4 of the total tickets to distribution, then I invite them to..

this is where it gets confusing because the odds do not seem correct when considering the winners were within 1/4 of

the tickets printed/released.

Just like the instants the raffle winners can go anywhere and be at any time.....it doesn't throw off the overall odds but

seems like it does....no matter at what point that top ticket is sold, the figures are still 1/125,000 if they're selling

500,000 tickets....this time around they sold 470,000+ tickets. A dent in the calculations, or I should say a ding.

Good luck with your debate...hopefully, you can find a less life threatening topic to discuss. This one will drive you nuts.

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GiveFive

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Jan 6, 2016, 6:54 am

Here's something else I'm curious about regarding the raffle.

Can a player win a prize twice with the same ticket number?

I would think that once ticket number, "00123456" had won a prize, it would no longer be eligible to win another one, ie that it would be removed from the "prize pool" of 500,000 possible winners. Anybody know?

And... are the first 4 numbers "drawn" (and I'm using that term loosely) the top prize winners? I guess there's a lot I don't know about the raffle! G5

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JADELottery

Native American Eagle Sun

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Jan 6, 2016, 10:34 am

Assuming all raffle tickets are sold and one raffle tick is purchased, then the Odds for winning the 4 Top Prizes are:

1st Top Prize Odds 1:499,999

2nd Top Prize Odds 1:499,998

3rd Top Prize Odds 1:499,997

4th Top Prize Odds 1:499,996

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dr65

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Jan 6, 2016, 4:58 pm

Jade, what are the odds when the tickets sell only to the quarter point and the 4 top winners
are in that first 125,000 but they sell 375,000 tickets out of 500,000?

I think your answer above has made at least 2 posters happy....now the OP still has the task
explaining it.

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JADELottery

Native American Eagle Sun

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Jan 6, 2016, 5:58 pm

Quote: Originally posted by dr65 on Jan 6, 2016
Jade, what are the odds when the tickets sell only to the quarter point and the 4 top winners
are in that first 125,000 but they sell 375,000 tickets out of 500,000?

I think your answer above has made at least 2 posters happy....now the OP still has the task
explaining it.

If the raffle only sold 375,000 tickets and you only purchased 1 ticket, then the odds for winning the top 4 prizes are:

1st Top Prize Odds 1:374,999

2nd Top Prize Odds 1:374,998

3rd Top Prize Odds 1:374,997

4th Top Prize Odds 1:374,996

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GiveFive

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Jan 15, 2016, 5:54 pm

Quote: Originally posted by JADELottery on Jan 6, 2016
Assuming all raffle tickets are sold and one raffle tick is purchased, then the Odds for winning the 4 Top Prizes are:

1st Top Prize Odds 1:499,999

2nd Top Prize Odds 1:499,998

3rd Top Prize Odds 1:499,997

4th Top Prize Odds 1:499,996

To me, the above explanation makes the most sense.

The first four unique eight-digit numbers drawn will be the first-prize-tier winning numbers. A player may win only one time on each ticket. So that means after each one is drawn, it is no longer eligible to win any other prize.

Sure, from a mathematical calculations stand point, 4/500,000 yields the same results as 1/125,000. But, assuming all 500,000 tickets are sold, after the first ticket number has been drawn and is removed from the pool of eligible ticket numbers, (and that number was NOT your ticket number) 499,999 tickets remain in the ticket pool, and they each are eligible to win 1 million dollars. Therefore, your ticket number is 1 of the 499,999 remaining eligible tickets. (1 in 499,999 odds) G5

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GiveFive

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Jan 15, 2016, 6:50 pm

Quote: Originally posted by GiveFive on Jan 15, 2016
To me, the above explanation makes the most sense.

The first four unique eight-digit numbers drawn will be the first-prize-tier winning numbers. A player may win only one time on each ticket. So that means after each one is drawn, it is no longer eligible to win any other prize.

Sure, from a mathematical calculations stand point, 4/500,000 yields the same results as 1/125,000. But, assuming all 500,000 tickets are sold, after the first ticket number has been drawn and is removed from the pool of eligible ticket numbers, (and that number was NOT your ticket number) 499,999 tickets remain in the ticket pool, and they each are eligible to win 1 million dollars. Therefore, your ticket number is 1 of the 499,999 remaining eligible tickets. (1 in 499,999 odds) G5

What are "Odds"? They're simply a tool used to measure the probability of something either happening or not happening.

In the case of playing the lottery, they're when the probability that the event will not happen is greater than the probability that it will, then the odds are said to be "against" that event happening.

Players don't buy odds. They buy tickets. See, I have trouble with the PA lottery advertising the odds for the Millionaire raffle being 125,000 to 1 (against the player) Yes, mathematically they are being truthful, so that's why they can put odds of 125,000 to 1 in their advertisements.

But the fact remains after the first ticket is drawn, there are 499,999 ticket numbers remaining that could be drawn. And if I bought 1 ticket, then my ticket number is 1 of those 499,999 numbers, and the chances of my ticket number being drawn is 1 for me, and 499,999 against me. G5

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JADELottery

Native American Eagle Sun

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Jan 16, 2016, 12:53 pm

Before we get into the logic of why we say the odds are as stated, we need to understand the truest sense of what a raffle is.

Raffles are similar to Lotteries in the fact they are a game of probability.

However, that's where the similarity ends.

Raffles differ from Lotteries because they guarantee a prize will be won by someone at the time of selection, Lotteries do not.

Typical manual raffles use a ticketing system where one buys a dual ticket stamped with identical numbers; seen below.

One half of the ticket is given to the buyer and the other half is used by the raffle proprietor to select a winner.

Those tickets are usually put in a mixing drum, mechanically spun; a winning ticket is randomly selected by hand and read.

The thing to note here is the selection process is sequential, meaning thickets are drawn one at a time.

Additional winners are determined by spinning the drum again and selecting another ticket for other prizes.

Even if these tickets were grabbed many at a time, each ticket would have to be read one at a time; making it sequential.

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JADELottery

Native American Eagle Sun

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Jan 16, 2016, 1:16 pm

The sequential nature of a raffle is verified by the PA Lottery's own Millionaire Raffle #23 rules.

From the PA State Lottery Law,

"

7. Conduct of Drawing: The results of the Pennsylvania Millionaire Raffle XXII lottery game will be posted to the Lottery's publicly accessible website on January 2, 2016, at or after 10:00 p.m. A computer-generated randomizer will be used to conduct the drawing. Six-thousand (6,000) unique eight-digit numbers will be drawn from the range of numbers representing the chances sold. The first four unique eight-digit numbers drawn will be the first-prize-tier winning numbers. The fifth through eighth unique eight-digit numbers drawn will be the second-prize-tier winning numbers. The ninth through 108th unique eight-digit numbers drawn will be the third-prize-tier winning numbers. The 109th through 6,000th unique eight-digit numbers drawn will be the fourth-prize-tier winning numbers.

"

Now, whether the winning ticket number is read in seconds or microseconds does not diminish the fact each winning prize number is drawn sequentially.

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JADELottery

Native American Eagle Sun

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Jan 16, 2016, 1:33 pm

Let's look at the manual aspect of a raffle.

If we assume there are 500,000 raffle tickets sold and each prize winning ticket is drawn one at a time from the mixing drum, then we need to examine what is happening at each moment of selection.

In the 1st Top Prize, one ticket is selected from the drum of 500,000.

The odds of one purchased ticket winning the 1st Top Prize is 1 : 499,999.

Now, the drum is closed and the tickets are mixed again.

How many tickets a left in the drum?

Yep, 499,999 left.

This means when the 2nd Top Prize is selected, the odds only drop to 1 : 499,998.

Next, the 3rd Top Prize odds would drop by 1 to 1 : 499,997.

As you can see in this mechanical raffle, the odds don't drastically change.

In the case of the electronic raffle, the odds don't change just because you're using a computer to do it.

It's the same whether drawn and read in seconds or microseconds.

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GiveFive

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Jan 16, 2016, 2:13 pm

Quote: Originally posted by JADELottery on Jan 16, 2016
Let's look at the manual aspect of a raffle.

If we assume there are 500,000 raffle tickets sold and each prize winning ticket is drawn one at a time from the mixing drum, then we need to examine what is happening at each moment of selection.

In the 1st Top Prize, one ticket is selected from the drum of 500,000.

The odds of one purchased ticket winning the 1st Top Prize is 1 : 499,999.

Now, the drum is closed and the tickets are mixed again.

How many tickets a left in the drum?

Yep, 499,999 left.

This means when the 2nd Top Prize is selected, the odds only drop to 1 : 499,998.

Next, the 3rd Top Prize odds would drop by 1 to 1 : 499,997.

As you can see in this mechanical raffle, the odds don't drastically change.

In the case of the electronic raffle, the odds don't change just because you're using a computer to do it.

It's the same whether drawn and read in seconds or microseconds.

Couldn't agree more.

Yes, it's a raffle, so 4 out of 500,000 tickets are guaranteed to win $1,000,000. So that's 4/500,000 or 0.000002%. And if you reduce 4/500,000 to it's least common denominator of 1/125,000 that's also 0.000002%. So technically, the odds can be stated as 1/125,000. And that's why the PA Lottery can advertise odds of 1/125,000.

But as can be seen in JADE's posts above, the odds are really 1 : 499,999, 1 : 499,998 etc. But if they advertised the odds in that particular fashion, that wouldn't help ticket sales as much as 1/125,000 will. G5

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JADELottery

Native American Eagle Sun

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Jan 16, 2016, 2:26 pm

Quote: Originally posted by GiveFive on Jan 16, 2016
Couldn't agree more.

Yes, it's a raffle, so 4 out of 500,000 tickets are guaranteed to win $1,000,000. So that's 4/500,000 or 0.000002%. And if you reduce 4/500,000 to it's least common denominator of 1/125,000 that's also 0.000002%. So technically, the odds can be stated as 1/125,000. And that's why the PA Lottery can advertise odds of 1/125,000.

But as can be seen in JADE's posts above, the odds are really 1 : 499,999, 1 : 499,998 etc. But if they advertised the odds in that particular fashion, that wouldn't help ticket sales as much as 1/125,000 will. G5

actaully, they don't even use the word 'Odds', they use 'Chance'.

from the PA Lottery website:

Chance is an undefined mathematical philosophical pseudonym of probability.

Real probability has mathematical constructs like: probability of success, probability of failure, odds and their interwoven computational relationships.

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