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Powerball has a special feature that gives players the opportunity to increase the size of their winnings. By purchasing the Power Play option for $1 more, all prizes except the Jackpot will be increased by a multiplier. Beginning January 2009, the second place tier of 5 white balls will always be multiplied by a factor of 5, while the other lower level tiers wil be multiplied by either 2, 3, 4 or 5. The multiplier will be selected from a pool of 16 numbers just prior to the Powerball draw. The pool contains (4) 2's,(4) 3's, (4) 4's, and (4) 5's.
The average lower tier PowerPlay multiplier is 3.5, which is used for calculation purposes. This is determined by taking the probability weighted average of the four possible multipliers: 0.25 * ( 2 + 3 + 4 + 5 ) = 3.5
Is this a good value for players? Should the player buy this option? Lottery Power Picks believes that the answers are both Yes and No , depending on the Jackpot size.
For this analysis, LotteryPowerPicks calculated and compared the the payout return values of buying 1 Powerball Ticket against buying 1 ticket with the Power Play Option for $2. The higher the return, the better for players.
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Table PB-PP1 Powerball Summary
Money Returned to Players per $1 Spent
| | Jackpot | $0 | $20,000,000 | $62,568,435 | $120,000,000
| | 1 Ticket | $0.175 | $0.277 | $0.495 | $0.789
| w/ Power Play | $0.335 | $0.386 | $0.495 | $0.642
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| Breakeven
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As Table PB-PP1 illustrates, the Power Play Option returns more money to players when theJackpot is less than $62,568,435. When the Jackpot is greater than this amount, more money is returned to players who buy tickets without the Power Play Option.
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Powerball Power Play Rule
Jackpot < $62,568,435: Buy Power Play
Jackpot > $62,568,435: Do NOT Buy Power Play
Introducing the Multiplier Jackpot Breakeven Formula
J = O*(m-1.6658096)*c
J = Breakeven Jackpot Amount - Calculated
O = Odds of Winning Jackpot = 195,249,054 given
m = multiplier = 3.50 Calculated (see above)
c = constant ( sum( p(i)*prize(i) where i=2,n ) = 0.175 from Table PB-PP3a
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How These are Calculated
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| Probability
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| $0 Jackpot
Constant Variable 'c'
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| $20 Million Jackpot
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| $62,568,435 Million
Jackpot Breakeven
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| $120 Million Jackpot
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Table PB-PP2 Powerball
Probability Matrix
One Ticket
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| Won | Probability
| | 5+mb | 0.000001%
| | 5 | 0.000019%
| | 4+mb | 0.000138%
| | 4 | 0.005255%
| | 3+mb | 0.007329%
| | 3 | 0.278506%
| | 2+mb | 0.127038%
| | 2 | 4.827434%
| | 1+mb | 0.809866%
| | 1 | 30.774894%
| | 0+mb | 1.619731%
| | 0 | 61.549789%
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Table PB-PP3a Powerball
$1 Single Ticket
No Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 0 | $0.000
| | 5 | 200,000 | $0.039
| | 4+mb | 10,000 | $0.014
| | 4 | 100 | $0.005
| | 3+mb | 100 | $0.007
| | 3 | 7 | $0.019
| | 2+mb | 7 | $0.009
| | 2 | | $0.000
| | 1+mb | 4 | $0.032
| | 1 | | $0.000
| | 0+mb | 3 | $0.049
| | 0 | | $0.000
| | Per $ | $0.175
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Constant Variable 'c'
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Table PB-PP3b Powerball
$1 Single Ticket
No Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 20,000,000 | $0.102
| | 5 | 200,000 | $0.039
| | 4+mb | 10,000 | $0.014
| | 4 | 100 | $0.005
| | 3+mb | 100 | $0.007
| | 3 | 7 | $0.019
| | 2+mb | 7 | $0.009
| | 2 | | $0.000
| | 1+mb | 4 | $0.032
| | 1 | | $0.000
| | 0+mb | 3 | $0.049
| | 0 | | $0.000
| | Per $ | $0.277
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Table PB-PP3c Powerball
$1 Single Ticket
No Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 62,568,435 | $0.320
| | 5 | 200,000 | $0.039
| | 4+mb | 10,000 | $0.014
| | 4 | 100 | $0.005
| | 3+mb | 100 | $0.007
| | 3 | 7 | $0.019
| | 2+mb | 7 | $0.009
| | 2 | | $0.000
| | 1+mb | 4 | $0.032
| | 1 | | $0.000
| | 0+mb | 3 | $0.049
| | 0 | | $0.000
| | Per $ | $0.495
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Jackpot Breakeven
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Table PB-PP3d Powerball
$1 Single Ticket
No Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 120,000,000 | $0.615
| | 5 | 200,000 | $0.039
| | 4+mb | 10,000 | $0.014
| | 4 | 100 | $0.005
| | 3+mb | 100 | $0.007
| | 3 | 7 | $0.019
| | 2+mb | 7 | $0.009
| | 2 | | $0.000
| | 1+mb | 4 | $0.032
| | 1 | | $0.000
| | 0+mb | 3 | $0.049
| | 0 | | $0.000
| | Per $ | $0.789
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| With Power Play Option
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Table PB-PP4a Powerball
$2 Single Ticket
With 3.5x Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 0 | $0.000
| | 5 | 1,000,000 | $0.195
| | 4+mb | 35,000 | $0.048
| | 4 | 350 | $0.018
| | 3+mb | 350 | $0.026
| | 3 | 24.50 | $0.068
| | 2+mb | 24.50 | $0.031
| | 2 | | $0.000
| | 1+mb | 14 | $0.113
| | 1 | | $0.000
| | 0+mb | 10.5 | $0.170
| | 0 | | $0.000
| | Sum | $0.670
| | Per $ | $0.335
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Table PB-PP4b Powerball
$2 Single Ticket
With 3.5x Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 20,000,000 | $0.102
| | 5 | 1,000,000 | $0.195
| | 4+mb | 35,000 | $0.048
| | 4 | 350 | $0.018
| | 3+mb | 350 | $0.026
| | 3 | 24.50 | $0.068
| | 2+mb | 24.50 | $0.031
| | 2 | | $0.000
| | 1+mb | 14 | $0.113
| | 1 | | $0.000
| | 0+mb | 10.5 | $0.170
| | 0 | | $0.000
| | Sum | $0.772
| | Per $ | $0.386
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Table PB-PP4c Powerball
$2 Single Ticket
With 3.5x Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 62,568,435 | $0.320
| | 5 | 1,000,000 | $0.195
| | 4+mb | 35,000 | $0.048
| | 4 | 350 | $0.018
| | 3+mb | 350 | $0.026
| | 3 | 24.50 | $0.068
| | 2+mb | 24.50 | $0.031
| | 2 | | $0.000
| | 1+mb | 14 | $0.113
| | 1 | | $0.000
| | 0+mb | 10.5 | $0.170
| | 0 | | $0.000
| | Sum | $0.990
| | Per $ | $0.495
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Jackpot Breakeven
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Table PB-PP4d Powerball
$2 Single Ticket
With 3.5x Power Play
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| Won | Prize | Prob
Wtd
| | 5+mb | 120,000,000 | $0.615
| | 5 | 1,000,000 | $0.195
| | 4+mb | 35,000 | $0.048
| | 4 | 350 | $0.018
| | 3+mb | 350 | $0.026
| | 3 | 24.50 | $0.068
| | 2+mb | 24.50 | $0.031
| | 2 | | $0.000
| | 1+mb | 14 | $0.113
| | 1 | | $0.000
| | 0+mb | 10.5 | $0.170
| | 0 | | $0.000
| | Sum | $1.284
| | Per $ | $0.642
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Calculating the Powerball Breakeven Jackpot
J = O*(m-2)*c
J = 195,249,054*(3.5-1.6658096)*0.175
J = $62,568,435
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