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Rollover probabilities for Powerball at $400M annuity.

Topic closed. 10 replies. Last post 3 years ago by Stack47.

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New Jersey
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Posted: September 15, 2013, 9:17 am - IP Logged

The Powerball jackpot is advertised as a 400M annuity jackpot.   This means, based on cash values of this jackpot and the previous jackpot that the lottery anticipates sales of $145.6M or about 72.8M tickets.   If these projections are accurate, the probability of various numbers of winners is as follows:

k, number of winnersp(m,k)
066.04%
127.40%
25.68%
30.79%
40.08%
50.01%

 The expectation value - with 1.00 being thought of as a "good bet" - is 0.75.

The most probable outcome at this point is another rollover.   This may change if the jackpot grows beyond the estimated values, and I will post updates if I can.

    Uff Da!'s avatar - InCelebration 001.jpg
    Washington State
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    Posted: September 15, 2013, 10:32 am - IP Logged

    Thanks for this information, Prob988.  I always find the probability of a rollover to be of interest.

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      Kentucky
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      Posted: September 15, 2013, 8:26 pm - IP Logged

      The Powerball jackpot is advertised as a 400M annuity jackpot.   This means, based on cash values of this jackpot and the previous jackpot that the lottery anticipates sales of $145.6M or about 72.8M tickets.   If these projections are accurate, the probability of various numbers of winners is as follows:

      k, number of winnersp(m,k)
      066.04%
      127.40%
      25.68%
      30.79%
      40.08%
      50.01%

       The expectation value - with 1.00 being thought of as a "good bet" - is 0.75.

      The most probable outcome at this point is another rollover.   This may change if the jackpot grows beyond the estimated values, and I will post updates if I can.

      If we use past ticket sales on $400 million jackpot as a starting point, sales should be at least $250 million or 125 million tickets.

        LottoMetro's avatar - Lottery-024.jpg
        Happyland
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        Posted: September 15, 2013, 8:33 pm - IP Logged

        MUSL projects sales of approximately 87 million tickets; this is likely a conservative projection but it will be updated Wednesday afternoon. At this figure just under 40% of all combinations will be sold. Even if sales hit 125 million tickets only 51% of combinations would be sold. Thinking in terms of cumulative probabilities, we are getting close to the 90% level.

        If the chances of winning the jackpot are so slim, why play when the jackpot is so small? Your chances never change, but the potential payoff does.
        If a crystal ball showed you the future of the rest of your life, and in that future you will never win a jackpot, would you still play?

        2017: 0% (0 tickets)
        P&L % = Total Win($)/Total Wager($) - 1

          Jon D's avatar - calotterylogo
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          Posted: September 15, 2013, 9:29 pm - IP Logged

          The Powerball jackpot is advertised as a 400M annuity jackpot.   This means, based on cash values of this jackpot and the previous jackpot that the lottery anticipates sales of $145.6M or about 72.8M tickets.   If these projections are accurate, the probability of various numbers of winners is as follows:

          k, number of winnersp(m,k)
          066.04%
          127.40%
          25.68%
          30.79%
          40.08%
          50.01%

           The expectation value - with 1.00 being thought of as a "good bet" - is 0.75.

          The most probable outcome at this point is another rollover.   This may change if the jackpot grows beyond the estimated values, and I will post updates if I can.

          One thing I'd caution you on is quoting an expectation value or EV for Jackpot games.

          There are many times where return/payout and EV can be used interchangably, but never for a single draw of a Jackpot game like MM or PB. The EV for the PB/MM return is 0.5 (or -50% or whatever) and does not change. You can talk about the theoretical return if you played all combinations with no other players winning for this draw, but that is not EV for the game.

          The top prize is pari-mutuel, based on sales and previous rollovers, and is split amongst multiple winners. You cannot choose to only look at a single draw or some kind of "instantaneous EV" which doesn't make sense, and ignoring other players and previous draws and contribution to the current jackpot. You could just as well pick any jackpot value out of the air for that matter.

          Not to mention that trying to say you have an expectation of 50c or 75c return on the dollar for your bet is really ludicrous for just about any person on earth. A more realistic return is 5-10% range for the average mortal.

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            Posted: September 16, 2013, 12:30 am - IP Logged

            One thing I'd caution you on is quoting an expectation value or EV for Jackpot games.

            There are many times where return/payout and EV can be used interchangably, but never for a single draw of a Jackpot game like MM or PB. The EV for the PB/MM return is 0.5 (or -50% or whatever) and does not change. You can talk about the theoretical return if you played all combinations with no other players winning for this draw, but that is not EV for the game.

            The top prize is pari-mutuel, based on sales and previous rollovers, and is split amongst multiple winners. You cannot choose to only look at a single draw or some kind of "instantaneous EV" which doesn't make sense, and ignoring other players and previous draws and contribution to the current jackpot. You could just as well pick any jackpot value out of the air for that matter.

            Not to mention that trying to say you have an expectation of 50c or 75c return on the dollar for your bet is really ludicrous for just about any person on earth. A more realistic return is 5-10% range for the average mortal.

            The expectation value as calculated includes values for all of the lower prizes, which amounts to about .18, as well as the weighted values calculated from the probabilities of numbers of winners based on projected sales, as calculated from a Poisson distribution assuming randomized tickets, as well as the advertised cash value and the odds of any particular ticket winning, which are of course constant.  In this case, the chief contribution is a single winner, which adds 0.51 to my calculated expectation value, followed by two winners, which adds 0.06 to the expectation value, in part because the probability of this occuring is lower, as is the size of the prize awarded to a winner.

            The $2.00 cost of a ticket suggests that Powerball will be hard pressed to produce a "good bet" of approximately 1.00 without also producing a record jackpot never before seen. 

            It actually produced jackpots that were "good bets" several times before with the $1.00 ticket, but my crude models suggest that to do this now, with a $2.00 ticket, it would need a jackpot on the order of a billion dollars annuity, depending on the path taken to the jackpot, i.e. the size of previous rollovers, and thus the number of tickets sold on the last draw.   This is not impossible, but it is also unlikely, although not as unlikely of winning the lottery itself.

            I find this type of calculation useful, and play accordingly, and share it "as is" but you are free to use other approaches.

              Jon D's avatar - calotterylogo
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              Posted: September 16, 2013, 9:39 am - IP Logged

              The expectation value as calculated includes values for all of the lower prizes, which amounts to about .18, as well as the weighted values calculated from the probabilities of numbers of winners based on projected sales, as calculated from a Poisson distribution assuming randomized tickets, as well as the advertised cash value and the odds of any particular ticket winning, which are of course constant.  In this case, the chief contribution is a single winner, which adds 0.51 to my calculated expectation value, followed by two winners, which adds 0.06 to the expectation value, in part because the probability of this occuring is lower, as is the size of the prize awarded to a winner.

              The $2.00 cost of a ticket suggests that Powerball will be hard pressed to produce a "good bet" of approximately 1.00 without also producing a record jackpot never before seen. 

              It actually produced jackpots that were "good bets" several times before with the $1.00 ticket, but my crude models suggest that to do this now, with a $2.00 ticket, it would need a jackpot on the order of a billion dollars annuity, depending on the path taken to the jackpot, i.e. the size of previous rollovers, and thus the number of tickets sold on the last draw.   This is not impossible, but it is also unlikely, although not as unlikely of winning the lottery itself.

              I find this type of calculation useful, and play accordingly, and share it "as is" but you are free to use other approaches.

              I just have this problem with ever calling MM or PB a "good bet" even when the jackpot is high, that's just my pet peeve.

              EV is more appropriate for low odds gambling like blackjack or craps, but really has no place with high odds jackpot games with pari-mutuel top prizes. But anyway, the return for this draw is either quite a bit lower than 0.75, or quite a bit higher if you include the wager itself for all combos. And it makes way too many assumptions that you win the jackpot with your bet, or you buy all combos and assume nobody else wins to top prize. And that's before any tax liability.

              The rollover possibilities are somewhat interesting, but as always, they have little bearing on any single draw. As you know, the 5+1 jackpot can be won with as little as 15M tickets or could take as much as 230M tickets for just a single winner. You never know.

              Below is a plot of number of winners as a function of ticket sales. (0, 1 or >=2 winners) This next draw 9/18/13 is targeted in the 80M ticket range:

              PowerballProbability012

              And each individually, similarly representing your poisson calc numbers:

              PowerballProbabilityAll

                RedStang's avatar - tallman zps6gf4inoc.jpg
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                Posted: September 16, 2013, 2:26 pm - IP Logged

                Roll one more time then someone will prevent it from breaking the MM record.

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                  Posted: September 16, 2013, 8:53 pm - IP Logged

                  I just have this problem with ever calling MM or PB a "good bet" even when the jackpot is high, that's just my pet peeve.

                  EV is more appropriate for low odds gambling like blackjack or craps, but really has no place with high odds jackpot games with pari-mutuel top prizes. But anyway, the return for this draw is either quite a bit lower than 0.75, or quite a bit higher if you include the wager itself for all combos. And it makes way too many assumptions that you win the jackpot with your bet, or you buy all combos and assume nobody else wins to top prize. And that's before any tax liability.

                  The rollover possibilities are somewhat interesting, but as always, they have little bearing on any single draw. As you know, the 5+1 jackpot can be won with as little as 15M tickets or could take as much as 230M tickets for just a single winner. You never know.

                  Below is a plot of number of winners as a function of ticket sales. (0, 1 or >=2 winners) This next draw 9/18/13 is targeted in the 80M ticket range:

                  PowerballProbability012

                  And each individually, similarly representing your poisson calc numbers:

                  PowerballProbabilityAll

                  There is no mathematically legitimate way to declare the expectation value as undefined because the odds are large.   Indeed in mathematical physics, one often defines improbable events in terms of expectation values, in the case of so called "quantum mechanical tunneling" for example. 

                  It is simply a ratio of potential reward to risk, and there is no arbitrary place at which it is undefined except where the risk is non-existent.  If the ratio of the sum of cash jackpots awarded to each winner times the probability of that number of winners is equal to the cost of the ticket times the odds of the ticket winning, the expectation value is 1.   Whether semantically one chooses to not define this as a "good bet" is a personal choice of course.   They often say that playing the lottery is not a game for people who know math, but I play at a certain point anyway, since I recognize that the universe is a sum of extremely improbable events, and its a fun fantasy to play with, and, in my state at least, for a good cause.

                  However, to each his own.   

                  As I said before, one can take it for what it's worth.   I was merely telling people on how I evaluate jackpots, and will continue to evaluate jackpots.   I play when the expectation value reaches a certain level.

                  Nice graphs though of the Poisson distribution though.   I never bothered to plug these into excel and plot them but they are nice representations and fun to look at.

                    Jon D's avatar - calotterylogo
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                    Posted: September 16, 2013, 10:05 pm - IP Logged

                    There is no mathematically legitimate way to declare the expectation value as undefined because the odds are large.   Indeed in mathematical physics, one often defines improbable events in terms of expectation values, in the case of so called "quantum mechanical tunneling" for example. 

                    It is simply a ratio of potential reward to risk, and there is no arbitrary place at which it is undefined except where the risk is non-existent.  If the ratio of the sum of cash jackpots awarded to each winner times the probability of that number of winners is equal to the cost of the ticket times the odds of the ticket winning, the expectation value is 1.   Whether semantically one chooses to not define this as a "good bet" is a personal choice of course.   They often say that playing the lottery is not a game for people who know math, but I play at a certain point anyway, since I recognize that the universe is a sum of extremely improbable events, and its a fun fantasy to play with, and, in my state at least, for a good cause.

                    However, to each his own.   

                    As I said before, one can take it for what it's worth.   I was merely telling people on how I evaluate jackpots, and will continue to evaluate jackpots.   I play when the expectation value reaches a certain level.

                    Nice graphs though of the Poisson distribution though.   I never bothered to plug these into excel and plot them but they are nice representations and fun to look at.

                    There is no mathematically legitimate way to declare the expectation value as undefined because the odds are large.

                    I didn't say the EV was undefined. As I said in my original reply: The EV for the PB/MM return is 0.5 and does not change. You wanna talk about the return for a single draw, that is one thing, but that is not the same as the EV for the game. The odds and prize share percentage do not change, and the rollovers and variable jackpots are just part of the game and the long term average.

                    I guess we will agree to disagree.

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                      Kentucky
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                      Posted: September 16, 2013, 10:44 pm - IP Logged

                      There is no mathematically legitimate way to declare the expectation value as undefined because the odds are large.

                      I didn't say the EV was undefined. As I said in my original reply: The EV for the PB/MM return is 0.5 and does not change. You wanna talk about the return for a single draw, that is one thing, but that is not the same as the EV for the game. The odds and prize share percentage do not change, and the rollovers and variable jackpots are just part of the game and the long term average.

                      I guess we will agree to disagree.

                      I use the simple approach by projecting the number of five number matches. It should require about $206 million in sales to produce 20 five number matches. Based on sales from past jackpots of this size, the total ticket sales could reach $250 million and around 25 five number matches. With each of them having a 1 in 35 chance of matching the bonus number, the chances of the jackpot being won should be over 50%. Based on projected sales of $145.6 million, there is less than a 50% chance.

                      I'm not saying it's the best scientific approach, but it's not that far off.