seven
seven
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Mission146
September 14th, 2021 at 8:30:56 AM permalink
hi @all

let's say a lottery participant has 10 tickets which costs him all in all $1 in a lottery game with 2100 tickets. There will be a draw with 2 winners who get $450 each.

What is the player's EV?

I am confused about the 2 winners of the draw

thank you as always in advance for your help

cheers
ThatDonGuy
ThatDonGuy
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Mission146
September 14th, 2021 at 8:41:28 AM permalink
There are C(2100, 2) = 2,203,950 pairs of winning tickets.
There are three possible results as far as the player is concerned:

1. There are 45 pairs where the player has both tickets; the ER (expected return) of this is 45 / 2,203,950 x 900

2. There are 10 x 2090 = 20,900 pairs where the player has one ticket; the ER is 20,900 / 2,203,950 x 450.

3. There are C(2090, 2) pairs where the player wins nothing.

The total ER is 9,445,000 / 2,203,950 = 30/7.
Since the player paid 10 for the tickets, the EV is (30/7 - 10) = -40/7.
seven
seven
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September 14th, 2021 at 10:07:55 AM permalink
Quote: ThatDonGuy

There are C(2100, 2) = 2,203,950 pairs of winning tickets.
There are three possible results as far as the player is concerned:

1. There are 45 pairs where the player has both tickets; the ER (expected return) of this is 45 / 2,203,950 x 900

2. There are 10 x 2090 = 20,900 pairs where the player has one ticket; the ER is 20,900 / 2,203,950 x 450.

3. There are C(2090, 2) pairs where the player wins nothing.

The total ER is 9,445,000 / 2,203,950 = 30/7.
Since the player paid 10 for the tickets, the EV is (30/7 - 10) = -40/7.

  • link to original post



    thank you very much for the answer but I need to apologize that I did not explain correctly/understandable. I am really sorry

    the player who has 10 tickets paid $1 for all the 10 tickets ( he did not pay $10 for the 10 tickets)

    the 2 winner prizes will go to 2 separate players means if one player wins he can not win the 2nd prize

    I hope I could clear it up now

    cheers

    edit
    maybe I should explain the 2100 tickets draw
    there will be 2100 tickets in a hat and one ticket will be drawn for the 1st winner then there will be drawn another ticket for the 2nd winner. if the 1st winner will win again it will not count and another ticket will be drawn
    Mission146
    Mission146 
    Joined: May 15, 2012
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    September 14th, 2021 at 10:32:06 AM permalink
    Quote: seven



    thank you very much for the answer but I need to apologize that I did not explain correctly/understandable. I am really sorry

    the player who has 10 tickets paid $1 for all the 10 tickets ( he did not pay $10 for the 10 tickets)

    the 2 winner prizes will go to 2 separate players means if one player wins he can not win the 2nd prize

    I hope I could clear it up now

    cheers

    edit
    maybe I should explain the 2100 tickets draw
    there will be 2100 tickets in a hat and one ticket will be drawn for the 1st winner then there will be drawn another ticket for the 2nd winner. if the 1st winner will win again it will not count and another ticket will be drawn

  • link to original post



    Okay, there are three possible combinations of events that can happen in this one. What we're going to do here is express them in the simplest terms:

    1.) The first ticket drawn results in the guy winning:

    10/2100 = 0.00476190476

    2.) He does not win on the first ticket, but then he does win on the second ticket drawn:

    (2090/2100) * (10/2099) = 0.00474148687

    3.) He does not win on either ticket drawn:

    (2090/2100) * (2089/2099) = 0.99049660836

    Okay, so we know that he either wins $450 (which is really $440 profits because he paid for the tickets) or that he loses $10. With that, we have:

    ((0.00476190476+0.00474148687)*440) - (10 * 0.99049660836) = -5.7234737664

    This is only slightly worse than what ThatDonGuy ended up with and the reason why is simply that the guy cannot win twice. It also assumes that the original $10 for the tickets is sunk (pays on a For One basis) even if he wins.
    https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
    seven
    seven
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    September 14th, 2021 at 10:49:46 AM permalink
    Quote: Mission146

    Quote: seven



    thank you very much for the answer but I need to apologize that I did not explain correctly/understandable. I am really sorry

    the player who has 10 tickets paid $1 for all the 10 tickets ( he did not pay $10 for the 10 tickets)

    the 2 winner prizes will go to 2 separate players means if one player wins he can not win the 2nd prize

    I hope I could clear it up now

    cheers

    edit
    maybe I should explain the 2100 tickets draw
    there will be 2100 tickets in a hat and one ticket will be drawn for the 1st winner then there will be drawn another ticket for the 2nd winner. if the 1st winner will win again it will not count and another ticket will be drawn

  • link to original post



    Okay, there are three possible combinations of events that can happen in this one. What we're going to do here is express them in the simplest terms:

    1.) The first ticket drawn results in the guy winning:

    10/2100 = 0.00476190476

    2.) He does not win on the first ticket, but then he does win on the second ticket drawn:

    (2090/2100) * (10/2099) = 0.00474148687

    3.) He does not win on either ticket drawn:

    (2090/2100) * (2089/2099) = 0.99049660836

    Okay, so we know that he either wins $450 (which is really $440 profits because he paid for the tickets) or that he loses $10. With that, we have:

    ((0.00476190476+0.00474148687)*440) - (10 * 0.99049660836) = -5.7234737664

    This is only slightly worse than what ThatDonGuy ended up with and the reason why is simply that the guy cannot win twice. It also assumes that the original $10 for the tickets is sunk (pays on a For One basis) even if he wins.
  • link to original post



    I don't understand why 3 possible combinations

    the player with 10 tickets in case one of his tickets is drawn out of the hat will win $450 minus $1 equals $449 profit. the winner ticket is out and does not go back into the hat.

    in case the next ticket which is drawn is also from the player who owns 10 tickets the ticket is out of the hat and another ticket is drawn until a 2nd winner is drawn

    what is the EV of the player with 10 tickets?
    Mission146
    Mission146 
    Joined: May 15, 2012
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    September 14th, 2021 at 11:19:48 AM permalink
    Quote: seven



    I don't understand why 3 possible combinations

    the player with 10 tickets in case one of his tickets is drawn out of the hat will win $450 minus $1 equals $449 profit. the winner ticket is out and does not go back into the hat.

    in case the next ticket which is drawn is also from the player who owns 10 tickets the ticket is out of the hat and another ticket is drawn until a 2nd winner is drawn

    what is the EV of the player with 10 tickets?

  • link to original post



    Your original post stipulated that he has ten tickets. Is he going to get a refund on the nine that don't win? If you want to know his EV, then it includes the fact that he has purchased ten tickets.

    Three combinations because:

    His ticket could be the first drawn: 10/2100 is the probability of that event. 0.00476190476

    His ticket could be the second drawn, which means some other ticket was drawn first: (2090/2100 * 10/2099) = 0.00474148687 (As you can see, this is slightly less likely than one of his tickets being the first drawn, mainly just because it includes another event happening)

    Both tickets drawn are none of his ten.
    https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
    Mission146
    Mission146 
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    September 14th, 2021 at 11:24:08 AM permalink
    Or, are you saying that the tickets cost a TOTAL of $1?

    If that's the case, just replace "440" and "10":

    ((0.00476190476+0.00474148687)*440) - (10 * 0.99049660836) =

    With "449" and "1", respectively.
    https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
    seven
    seven
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    September 14th, 2021 at 11:27:20 AM permalink
    Quote: Mission146

    Quote: seven



    I don't understand why 3 possible combinations

    the player with 10 tickets in case one of his tickets is drawn out of the hat will win $450 minus $1 equals $449 profit. the winner ticket is out and does not go back into the hat.

    in case the next ticket which is drawn is also from the player who owns 10 tickets the ticket is out of the hat and another ticket is drawn until a 2nd winner is drawn

    what is the EV of the player with 10 tickets?

  • link to original post



    Your original post stipulated that he has ten tickets. Is he going to get a refund on the nine that don't win? If you want to know his EV, then it includes the fact that he has purchased ten tickets.
  • link to original post



    sorry for the confusion. he purchased 1 ticket for $1 and got another 9 for free so he has 10 tickets for $1 and he did not pay $10 for ten tickets. he does not get any refund.
    as my confusion was because of the 2 winner option maybe we try the simple option with one prize winner of $450. in the hat are 2100 tickets and there will be drawn 1 ticket and the winner gets $450. what is the EV for a player with 10 tickets he paid $1 for all ten tickets ( he did not pay $10 for the ten tickets)?
    Mission146
    Mission146 
    Joined: May 15, 2012
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    Thanks for this post from:
    seven
    September 14th, 2021 at 11:47:46 AM permalink
    Quote: seven



    sorry for the confusion. he purchased 1 ticket for $1 and got another 9 for free so he has 10 tickets for $1 and he did not pay $10 for ten tickets. he does not get any refund.
    as my confusion was because of the 2 winner option maybe we try the simple option with one prize winner of $450. in the hat are 2100 tickets and there will be drawn 1 ticket and the winner gets $450. what is the EV for a player with 10 tickets he paid $1 for all ten tickets ( he did not pay $10 for the ten tickets)?

  • link to original post



    (449 * (10/2100)) - (1 * (2090/2100) = 1.14285714286 (This assumes only one winner of $450)

    ((0.00476190476+0.00474148687)*449) - (1 * 0.99049660836) = 3.27652623351 (This is ten tickets with two winners of $450, but the same person cannot win twice)
    https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
    seven
    seven
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    September 14th, 2021 at 11:10:01 PM permalink
    Quote: Mission146

    Quote: seven



    sorry for the confusion. he purchased 1 ticket for $1 and got another 9 for free so he has 10 tickets for $1 and he did not pay $10 for ten tickets. he does not get any refund.
    as my confusion was because of the 2 winner option maybe we try the simple option with one prize winner of $450. in the hat are 2100 tickets and there will be drawn 1 ticket and the winner gets $450. what is the EV for a player with 10 tickets he paid $1 for all ten tickets ( he did not pay $10 for the ten tickets)?

  • link to original post



    (449 * (10/2100)) - (1 * (2090/2100) = 1.14285714286 (This assumes only one winner of $450)

    ((0.00476190476+0.00474148687)*449) - (1 * 0.99049660836) = 3.27652623351 (This is ten tickets with two winners of $450, but the same person cannot win twice)
  • link to original post



    thank you very much for the formula as it looks like it answers my question but please be so kind if you can give it to me ELI5
    for example I know that a single zero roulette has a - 0.027 EV for the player means 2.7% house advantage
    what is the EV for the Player in % of your result 1.14285714286 (This assumes only one winner of $450)
    and 3.27652623351

    cheers

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