Probability of Sharing the Powerball Jackpot
Odds of a Jackpot Rollover, Odds of Splitting the Jackpot, Expected Share of the Jackpot
A common question among avid Powerball lottery players is what the odds are that they will have to share the jackpot if they win. And if the jackpot is likely to be split among more than one winning ticket, what the expected share per winner will be. Related to these questions are how likely it is that the Powerball jackpot will roll over to the next drawing if there is no winner. The answers to these questions help some players decide if it is worth buying a ticket, or if they should buy multiple tickets.
The odds in question depend on the number of lottery tickets sold and the probability of winning the jackpot with a single ticket. Here we answer all these questions using the Binomial Distribution, which is the probability distribution that models sequences or clusters of independent events, such as lotteries.
Binomial Theorem for Number of Powerball Lottery Jackpot Winners
If N is the number of Powerball tickets sold and p = 1/175223510 is the probability of winning, then the probability that there will be k winners is given by the binomial probability equation
B(N, p, k) = (N choose k) * p^k * (1p)^(Nk)
For example, if the lottery sells 165,000,000 tickets for a drawing, then the probability that there will be 0 winners is
(165000000 choose 0) * (1/175223510)^0 * (175223509/175223510)^165000000
≈ 0.39 = 39%
This is the same thing as the probability of a jackpot rollover. The more tickets are sold, the less likely a rollover becomes. The probability there will be exactly two winning Powerball tickets sold is
(165000000 choose 2) * (1/175223510)^2 * (175223509/175223510)^164999998
≈ 0.173 = 17.3%
Below is a table that summarizes the probabilities of the number of winners for various amounts of tickets sold, incremented in 25,000,000.
# of Tix Sold in Millions
 Prob. of No Winners
 Prob. of 1 Winner
 Prob. of 2 Winners
 Prob. of 3 Winners
 Prob. of 4 Winners
 Prob. of 5+ Winners


25
 0.867
 0.124
 0.009
 0.000
 0.000
 0.000

50
 0.752
 0.215
 0.030
 0.003
 0.000
 0.000

75
 0.652
 0.279
 0.060
 0.009
 0.001
 0.000

100
 0.565
 0.323
 0.092
 0.018
 0.002
 0.000

125
 0.490
 0.350
 0.125
 0.030
 0.005
 0.001

150
 0.425
 0.364
 0.156
 0.044
 0.010
 0.001

175
 0.368
 0.368
 0.184
 0.061
 0.015
 0.004

200
 0.319
 0.365
 0.208
 0.079
 0.023
 0.006

225
 0.277
 0.356
 0.228
 0.098
 0.032
 0.009

250
 0.240
 0.343
 0.244
 0.116
 0.041
 0.016

275
 0.208
 0.327
 0.256
 0.134
 0.053
 0.022

300
 0.180
 0.309
 0.266
 0.151
 0.065
 0.029

325
 0.156
 0.290
 0.269
 0.167
 0.077
 0.041

350
 0.136
 0.271
 0.271
 0.180
 0.090
 0.052

Notice that when there are approximately 175 million tickets sold, the probabilities of having no winners and one winner are about equal, and are the most likely outcomes. It's no coincidence that the total number of possible Powerball ticket number combinations is also about 175 million. In the same vein, when the total number of tickets sold is ~350 million, or about twice the total number of Powerball combinations, the probabilities of one and two winners are also about equal and are also the most likely outcomes.
Expected Number of Winners According to the Binomial Theorem
For a binomial distribution with parameters N and p, the expected value is N*p. In the case of the Powerball lottery, if there are N tickets sold and each ticket has a probability of p = 1/175223510 of being the winner, then the expected number of winners is
N*p = N/175223510
For example, if 240 million tickets are sold for a Powerball drawing, then the expected number of winners for that drawing is 240000000/175223510 = 1.37
When enough tickets have been sold so that the expected number of winners is at least 1, a crude way to estimate the winner's/winners' share of the jackpot is to divide the jackpot by N*p. For example, if the jackpot has reached $570 million because of previous rollovers and there were 215 million tickets sold for the current drawing, then the rough expected share per winner is
$570000000/(215000000/175223510)
≈ $464.5 million
However, there is a more rigorous way to compute the expected share of the jackpot per winner.
How to Calculate Estimated Share of Powerball Jackpot for N Tickets Sold
Let N be the number of tickets sold for a particular drawing of the Powerball lottery, let J be the amount of the jackpot, and let {P_{0}, P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, ...} be the probabilities that 0 winning tickets are sold, 1 winning ticket is sold, 2 winning tickets are sold, etc. The P_{i}'s are calculated using the Binomial Theorem above. On the condition that the jackpot does not rollover, i.e., there is at least one winner, then the expected share of the jackpot for each winner is given by the formula
Expected $ = J*[P_{1}/1 + P_{2}/2 + P_{3}/3 + P_{4}/4 + P_{5}/5 + ...] / [1  P_{0}]_{}
The Pi's range from P_{0} to P_{N}, but since N is a large number these probabilities rapidly decrease 0 after a certain point. It is therefore acceptable to use the truncated formula
Expected $ = J*[P_{1} + P_{2}/2 + P_{3}/3 + P_{4}/4 + P_{5}/5] / [1  P_{0}]
Example Calculation: Suppose the Powerball lottery sells 215 million tickets for a drawing and the jackpot is $570 million. Using the binomial formula,he probabilites of there being 0, 1, 2, 3, 4, and 5 winners are
P_{0} = 0.29317
P_{1} = 0.35972
P_{2} = 0.22069
P_{3} = 0.09026
P_{4} = 0.02769
P_{5} = 0.00679
Applying the formula, the expected amount of each winner's share is
570[0.35972+ 0.22069/2 +0.09026/3 +0.02769/4 +0.00679/5]/[1  0.29317]
≈ $410 million
Notice that this is lower than the crude formula estimate of $464.5 million obtained in the previous section.
How to Use These Figures
You can use these odds to calculate the expected return of a lottery ticket, or to decide when it's worth buying a ticket. The Powerball lottery is never a good bet or good investment, however it is theoretically possible for the expected return of a ticket to exceed the price of the ticket. This can only occur when there are many, many, many rollovers.
Some people only play Powerball when the number of tickets sold is at least 125 million or so, when the probability of there being no winner drops below 50%, i.e., it is more likely for there to be one or more winners than no winners.
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