100xOdds
100xOdds
Joined: Feb 5, 2012
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March 23rd, 2021 at 7:30:05 PM permalink
This state lottery changed the format for their Cash 5 game:
https://www.palottery.state.pa.us/Draw-Games/Cash-5/Prizes-Chances.aspx

Used to be $1 for a ticket thus if the jackpot was over $962,598, then it's +EV, assuming you're the only winner. (For this exercise, ignore taxes)

Now it's $2 for a ticket and they added a side game gimmick called 'Quick Cash' for the extra dollar.
Overall odds of winning that is 1:4.71

1/4.71 = .21
1-.21= .79

.79 x 962,598 = 760,452
962,598 + 760,452 = 1,723,050

so if the jackpot is greater than $1,723,050 then it's +EV? (again assuming you're the only winner and ignore taxes)

1) If not, then what's the answer?
2) And where did my math go wrong?
Craps is paradise (Pair of dice). Lets hear it for the SpeedCount Mathletes :)
Wizard
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Wizard
Joined: Oct 14, 2009
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March 24th, 2021 at 7:19:34 AM permalink
I get a breakeven jackpot of $1,190,941.95*.

That was a poorly written rules page you linked to, in my opinion. Here is some more needed to calculate the odds:

1. 43 total balls. Player picks 5.
2. As you mention, the ticket cost is $2.
3. Game pays jackpot for matching 5, $200 for matching 4, $10 for matching 3, and $2 for matching 2.
4. I assume the player gets something like a scratch-off with the pick-5 game. This scratch-off has a 1/80 chance of a win of $6 and 1/5 chance of a win of $2.

* Corrected
Last edited by: Wizard on Mar 24, 2021
It's not whether you win or lose; it's whether or not you had a good bet.
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
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March 24th, 2021 at 7:47:17 AM permalink
Quote: Wizard

I get a breakeven jackpot of $558,241.95.


How do you get this?
Here's what I get:
Each $2 ticket has two components - a Cash 5 component, and a Quick Cash component.
The Quick Cash component has an EV of 1/5 x 2 + 1/80 x 6 = 19/40
The Cash 5 component's EV is the sum of:
962,598
2 numbers: 2 x 84,360 / 962,598 = 168,720 / 962,598
3 numbers: 10 x 7030 / 962,598 = 70,300 / 962,598
4 numbers: 200 x 190 / 962,598 = 38,000 / 962,598
5 numbers: 1 / 962,598 x (the size of the jackpot) / (the expected number of jackpot winners)
If there are N tickets sold, the expected number of jackpot winners is N / 962,598, so the 5 numbers EV value = 1 / 962,598 x J / (N / 962,598) = J / N.
The total EV = 19/40 + 277,020 / 962,598 + jackpot size / number of tickets
According to the rules, the jackpot size = the previous jackpot size + N x 0.390632 (44.39% of each ticket's price goes into the prize pool, of which 44% goes into the jackpot)
The total EV = 1.1534157 + the previous jackpot amount / the number of tickets bought
Wizard
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Wizard
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March 24th, 2021 at 9:06:07 AM permalink
In my original math I put a win of $100 for matching 4. It is supposed to be only $10. This causes me to change my answer is $1,190,941.95.
Last edited by: Wizard on Mar 24, 2021
It's not whether you win or lose; it's whether or not you had a good bet.
100xOdds
100xOdds
Joined: Feb 5, 2012
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March 24th, 2021 at 6:38:51 PM permalink
Ahh.. didn't take the dollar amount into consideration when calculating the quick cash. (Thx)
19/40 = 47.5% return

.475 x 962,598 = 457,234

Taking only the Cash5 top prize into consideration (because that's the only prize people care about) then:
457,234 + 962,598 = $1,419,832

That's the point to be playing at?
Craps is paradise (Pair of dice). Lets hear it for the SpeedCount Mathletes :)
100xOdds
100xOdds
Joined: Feb 5, 2012
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March 27th, 2021 at 6:16:50 AM permalink
Here's another game (Match 6):
https://www.palottery.state.pa.us/Draw-Games/Match-6.aspx
https://www.palottery.state.pa.us/Draw-Games/Match-6.aspx#howtoplay

$2 per ticket, you get 3 plays. (so you get 18 numbers per $2 ticket)
odds of winning the top prize is 1 in 4,661,272 in a 3 play panel.

but they also have a secondary game gimmick.
if you match 4 to 10 numbers out of the 18, you also win some $.
the more #s you match, the more $ you get.

What's the +EV point of this game?
Thx
Craps is paradise (Pair of dice). Lets hear it for the SpeedCount Mathletes :)

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