Lottery question for you math types

[Smart_Cookie]

Canada's lottery organizations are currently running a national on-line lottery game that is different from any others.

It's called Millionaire Life, the grand prize is $1 million per year for 25 years. Details here.

It is going to be based on one grand-prize winning number out of all tickets sold - similar to sweepstakes, etc. (odds depend on actual number of entries received)

So...the more tickets sold, the greater the odds, right?

I already have 8 "chances". One $20 ticket and one $5 ticket. 8 different numbers.

The numbers are: xxxxxx-xx. So, by my math reasoning, there are 99 million possible numbers (0 to 99,999,999).

So let's say everyone in Canada buys one. (35 million or so - give or take). So right now my chances are 8 in 35 million.

Now, my emotions are saying "Buy another ticket or two. You'll have more chances to win". But my rational side is saying, no way. Because that's just increasing the number of tickets in the drum, so to speak. It won't better my odds at all. I'll just have 15 chances in 35 million.

Because what if everyone else does the same thing? And now, there are 70 or 80 million tickets sold.

I'm really going to try and control myself and NOT buy any more tickets.

Right?

p.s. and please don't tell me that I shouldn't have bought ANY tickets at all. I know the odds are long, but you have NO chances if you never buy a ticket!

 

[ChristineR]

Right now your chances are 8 in 35 million. Buy 8 more, and your chances will be 16 in 35,000,008, which is better.

Your decision to buy more tickets won't have much effect on whether or not other people buy tickets. If it's true that everyone thinks like you, then your chances will be 16 in 80,000,000, which is still a lot better than 8 in 79,999,992.

Are these tickets unique? What if they sell more than 100,000,000 tickets? Will they start duplicating numbers? Most lotto games pick numbers at random and allow duplicates.

 

[Ben Tilly]

Canada's lottery organizations are currently running a national on-line lottery game that is different from any others.

It's called Millionaire Life, the grand prize is $1 million per year for 25 years. Details here.

It is going to be based on one grand-prize winning number out of all tickets sold - similar to sweepstakes, etc. (odds depend on actual number of entries received)

So...the more tickets sold, the greater the odds, right?

The worse the odds you mean.

Anyways my immediate reaction is, "What is this really worth?" Let's say that half of what you make goes to taxes. (Makes the numbers easy.) Let's say that your discount rate is 5%. So the promise of $1.05 next year is equivalent to having $1 today. (Take $1, stick it in the bank, wait a year...actually 5% is low for a discount rate. 10% is a more normal figure. But what the heck, this is just a parameter.) So your $1 million/year is worth $500,000 to you for this year, $500,000/1.05 for next year, $500,000/(1.05^2) for 2 years from now, etc.

That means that (does a quick calculation...) that the grand prize is equivalent to $6,899,320.90. Little smaller than you were expecting, huh?

I already have 8 "chances". One $20 ticket and one $5 ticket. 8 different numbers.

The numbers are: xxxxxx-xx. So, by my math reasoning, there are 99 million possible numbers (0 to 99,999,999).

So let's say everyone in Canada buys one. (35 million or so - give or take). So right now my chances are 8 in 35 million.

Now, my emotions are saying "Buy another ticket or two. You'll have more chances to win". But my rational side is saying, no way. Because that's just increasing the number of tickets in the drum, so to speak. It won't better my odds at all. I'll just have 15 chances in 35 million.

Because what if everyone else does the same thing? And now, there are 70 or 80 million tickets sold.

I'm really going to try and control myself and NOT buy any more tickets.

Right?

p.s. and please don't tell me that I shouldn't have bought ANY tickets at all. I know the odds are long, but you have NO chances if you never buy a ticket!

The cheapest you can get a selection for is with the $20 ticket, where you get 7 for $20 at a cost of about $2.86 per selection. If (plugs in numbers) more than 2.4 million selections are sold nationwide, that lottery ticket doesn't pay for itself after taxes.

Does that help you exercise some self-restraint?

Cheers,
Ben

 

[Dan O.]

If you visit the lottery corporation's web site you can get some of the facts needed to estimate the payoff[*].

The grand prize of $25M over 25 years could also be taken as a straight $17M. There are also 4 $1M and 20 $100K secondary prizes. This gives a total of $23M in big prizes (with no CA tax).

There are minor prizes of $1000 for matching 6 digits and $20 for matching 2 digits. We can value these at 0.1 cents and 20 cents per number. For your $20 ticket you will get back (on average) $1.40 in minor prizes.

We don't know how many numbers will be sold so we cannot calculate the payoff on the major prizes. But we can compute the break even point...

$23M/($20-$1.40) = about 1.2M $20 tickets.

If you are sure that there are fewer than 1.2 million fools and dreamers in Canada it's a great deal and you should buy all the tickets you can.

Given that there are about 32M Canadians today[**], you would have to have an very low fool rate of 1 in 27. Even if there were only 1 fool in Canada, the US has enough idiots that will buy up those tickets and pull the payoff below the break even point.

 

[Grimoire]

Anyways my immediate reaction is, "What is this really worth?" Let's say that half of what you make goes to taxes. (Makes the numbers easy.)

In Canada, lottery winnings are not taxed. Makes the numbers even easier!

 

[Cuddles]

Now, my emotions are saying "Buy another ticket or two. You'll have more chances to win". But my rational side is saying, no way. Because that's just increasing the number of tickets in the drum, so to speak. It won't better my odds at all. I'll just have 15 chances in 35 million.

Because what if everyone else does the same thing? And now, there are 70 or 80 million tickets sold.

Ah, the joys of game theory. This is quite a common problem, and it doesn't have a general answer. One of the original games studied was the Prisoners' Dillema. Two people are in prison and are talked to by the police. They are both told that if they say nothing they will face a 5 year jail term, but if they turn in the other person they will go free while the other gets 20 years. However, if they both turn each other in they both get 15 years. What is the best choice? This simple question basically generated the entire field of game theory, and variants have been analysed for the last few decades. The problem depends very much on the exact values of reward/punishment and on the risk participants are willing to assume.

In your case the choices are, keep one ticket and have a low chance of winning or buy more with a higher chance of winning. If everyone buys more then if you lose, you have lost more money, and even if you win, you have not won quite as much as you would have, since you spent more initially. The answer depends entirely on what you think other people will do. If you buy more tickets but other people don't, you are more likely to win. If you don't buy more but others do, you are less likely to win. If everyone buys more tickets, nothing changes except for the money you lose if you don't win. The question you have to ask yourself is "Do I feel lucky, punk?" Well, do ya?

 

[Freddy]

I don't know if this helps, but you are probably more like to get struck by lightning several times than you are to win this thing, even if you buy 1000 chances.

 

[Ben Tilly]

In Canada, lottery winnings are not taxed. Makes the numbers even easier!

Ah. Oh well.

They are in the US, and that's why I assumed they would be.

Cheers,
Ben

 

[Smart_Cookie]

I don't know if this helps, but you are probably more like to get struck by lightning several times than you are to win this thing, even if you buy 1000 chances.

but I never go outside! I'll NEVER get struck by lightning.


Thanks for your input everyone.

 

[Michael Redman]

I've never seen a lottery where the cost of a choice is dependent on how many you buy. Interesting idea. I assume the average purchaser is going to spend more this way.