Instant Scratch Lottery odds question

Suppose there is a scratch lottery, with 2,550,000 tickets printed.

There are 255 tickets that win $100, for odds of winning that being 1 in 10000.

How would you compute the odds of getting 1 $100 winning ticket, if you bought 10,000 tickets? It is obviously not 100%, but I'm not sure how to go about calculating it.

I'm mostly interested in the formula, given N tickets printed, and X winning tickets of a particular denomination, what is the probability of getting Y winners of that denomination from a set of I tickets.

The approximate equation is:

Chance of Win % = [1 - (1 - W/T)^P] * 100

T = How many lottery tickets will be sold
W = How many tickets sold will be winners
P = How many tickets you are planning to buy

= 1 - (.9999^10000) = 1 - .367861046 = 63.21%

I hope that the tickets sell for $.01 each.

There are other winning prize denominations too.

I'm trying to determine how to calculate the expected number of winners from each denomination given a set of I tickets.

Each ticket is worth $2, and they have 5 "Games" on them.

The table below shows all possible outcomes in the format:
Prize Odds(1 in) # of winners

Win in 1 game
$35,000 850,000.00 3
$500 38,636.36 66
$100 10,000.00 255
$50 5,000.00 510
$20 2,200.17 1,159
$10 718.31 3,550
$5 33.33 76,519
$3 12.50 203,964
$2 8.77 290,739
Win in 2 games
$102 ($100 + $2) 19,921.88 128
$30 ($20 + $10) 2,500.00 1,020
$5 ($3 + $2) 35.98 70,881
$4 ($2 + $2) 14.26 178,784
Win in 3 games
$75 ($50 + $20 + $5) 10,000.00 255
$10 ($5 + $3 + $2) 166.85 15,283

Based on the table, the expected number of winners from a set of I tickets, of the $75 ($50+$20+5) and $100 categories should be the same. I'm trying to determine what that number is.

The total expected payback is 60.96%. I'm interested in how many winners of each denomination could be expected in a set of I tickets.

So does your formula mean that from a set of 10,000 tickets, you would expect 0.63 $100 (and $75) winners?

Racer, you are playing Ontario's SLOTS OF CASH!

As for the number of EXPECTED winners, it is still the number of tickets bought / total number of tickets x prizes offered at that category.

But the odds of getting at least one EXPECTED winner in a particular denomination can still be characterized in that formula above.

Indeed I am. I like how quickly you identified that.

I have a tracking database of everything I've done, I'm curious to see how far ahead or behind of the expected outcome I am. I know I'm ahead of the game on $100 winners, was curious to see by how much.

Are you playing them too?

I suppose the odds then, of NOT having a prize of a particular category in a set of tickets is 1- your formula

ie. 37% chance of NOT finding a $100 winner in 10,000 tickets.

I live in Ontario, and keep tabs on the scratch lotteries, so I knew when I saw the payouts that you were playing a $2 ticket somewhere, and I guessed from the payouts that it was an Ontario ticket.

I play the lottery when the grand prize exceeds the expected value, about $25 M for 6/49 and $50 M for LottoMax.

Scratch tickets are very much a losing proposition... you are far better off making a drive to one of the local casinos and putting your money into a slot machine or better yet, on a table game.

Not to worry, I'm fully aware that scratch tickets are bad.

Because of the slot play these give you in the OLG Racetrack Slots, it makes them not completely terrible.

When I'm gambling for the sake of gambling, I'm absolutely playing table games.

As for the Lotto's (6/49 and LottoMax), I do the same, 649 when its over $27,600,000, and Max when its over $50M (which is STILL -EV, but less so.

How would one determine when the jackpot is greater than the expected value?

Essentially, you take the odds of winning the jackpot (1 in 13,983,000 for a standard 6/49 lottery) and multiply by the ticket value (in Ontario, thats $2). If the jackpot is greater than that, then in the long run, its +EV to play, because the rarity of winning is made up for by how much you win.

It's actually more complicated than that because you have to take all of the other prizes into account.