NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 5, 2016, 4:00 pm - IP Logged

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

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

Arizona United States Member #165073 March 24, 2015 220 Posts Offline

Posted: January 5, 2016, 8:07 pm - IP Logged

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.)

NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 5, 2016, 9:40 pm - IP Logged

Quote: Originally posted by Murgatroyd on January 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!

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 6, 2016, 6:54 am - IP Logged

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

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

Pennsylvania United States Member #74096 May 2, 2009 22860 Posts Offline

Posted: January 6, 2016, 4:58 pm - IP Logged

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.

West Concord, MN United States Member #21 December 7, 2001 3675 Posts Offline

Posted: January 6, 2016, 5:58 pm - IP Logged

Quote: Originally posted by dr65 on January 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

Presented 'AS IS' and for Entertainment Purposes Only. Any gain or loss is your responsibility. Use at your own risk.

Order is a Subset of Chaos Knowledge is Beyond Belief Wisdom is Not Censored Douglas Paul Smallish Jehocifer

NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 15, 2016, 5:54 pm - IP Logged

Quote: Originally posted by JADELottery on January 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

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 15, 2016, 6:50 pm - IP Logged

Quote: Originally posted by GiveFive on January 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 willnot 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

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

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.

Presented 'AS IS' and for Entertainment Purposes Only. Any gain or loss is your responsibility. Use at your own risk.

Order is a Subset of Chaos Knowledge is Beyond Belief Wisdom is Not Censored Douglas Paul Smallish Jehocifer

West Concord, MN United States Member #21 December 7, 2001 3675 Posts Offline

Posted: January 16, 2016, 1:33 pm - IP Logged

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.

Presented 'AS IS' and for Entertainment Purposes Only. Any gain or loss is your responsibility. Use at your own risk.

Order is a Subset of Chaos Knowledge is Beyond Belief Wisdom is Not Censored Douglas Paul Smallish Jehocifer

NY State United States Member #92609 June 10, 2010 3692 Posts Online

Posted: January 16, 2016, 2:13 pm - IP Logged

Quote: Originally posted by JADELottery on January 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

About playing the lottery -- You will lose more than you win. Until you hit a jackpot. Then everything changes!

West Concord, MN United States Member #21 December 7, 2001 3675 Posts Offline

Posted: January 16, 2016, 2:26 pm - IP Logged

Quote: Originally posted by GiveFive on January 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.

Presented 'AS IS' and for Entertainment Purposes Only. Any gain or loss is your responsibility. Use at your own risk.

Order is a Subset of Chaos Knowledge is Beyond Belief Wisdom is Not Censored Douglas Paul Smallish Jehocifer