New Powerball (lottery) Rules Will Cost You More

December 16, 2011
By

(This article was first published on BioStatMatt » R, and kindly contributed to R-bloggers)

The popular news are reporting [1,2,3,4,5] that the Multi-State Lottery Commission (MUSL) will change the rules for their lottery game Powerball, effective Jan. 15, 2012. I sent an email to the MUSL (at 8:00am Dec, 14th) asking for the new official rules, but haven't received a response yet (as of 10:30am Dec, 16th). Hence, these calculations are based on information from the popular news.The rule changes are summarized as follows:
  • cost to play increased to $2 from 1$
  • jackpot starts at $40M instead of $20M
  • second prize starts at $1M instead of $0.2M
  • number of red balls (powerballs) reduced to 34 from 39
  • prize for matching the red ball alone increased to $4 from $3
The MUSL is touting the changes using the slogan "MORE, BIGGER, BETTER: more millionaires, bigger starting jackpots, and better overall odds." However, as I demonstrate below, the average loss incurred by playing Powerball will increase by more than two times, from $0.46 to $1.09. That is, under the new rules, Powerball players can expect to lose $1.09 for every $2 ticket purchased. In the long run, this is equivalent to exchanging a $2 bill for a $1 bill every time Powerball is played.Note that these calculations are approximate because the jackpot is estimated using historical jackpots, and very generous because taxes are not deducted from winnings.The following R code computes the expected winnings (loss) under the old and new rules. You download the script or source the file from within R like so:
R> source("http://biostatmatt.com/R/powerball.R")
# jackpot amounts (as of 12/16/2011)
# http://www.lottostrategies.com/script/jackpot_history/draw_date/101
# http://www.usamega.com/powerball-jackpot.asp
jackpots <- read.csv('http://biostatmatt.com/csv/jackpot.csv')
average_jackpot <- mean(jackpots$PrizeMillions)

# for convenience
ch <- choose

# old rules
# choose 5 from 59 white balls, 1 from 39 red balls

# probabilities of winning
powerball_old <- function(w, r)
    ch(5,w)*ch(54,5-w)/ch(59,5)*ch(1,r)*ch(38,1-r)/ch(39,1)

# 9 ways to win
p_old <- vector("numeric", length=9)
p_old[1] <- powerball_old(5, 1) # five white + powerball_old
p_old[2] <- powerball_old(5, 0) # five white
p_old[3] <- powerball_old(4, 1) # four white + powerball_old
p_old[4] <- powerball_old(4, 0) # four white
p_old[5] <- powerball_old(3, 1) # three white + powerball_old
p_old[6] <- powerball_old(3, 0) # three white
p_old[7] <- powerball_old(2, 1) # two white + powerball_old
p_old[8] <- powerball_old(1, 1) # one white + powerball_old
p_old[9] <- powerball_old(0, 1) # powerball_old

# winnings
w_old <- vector("numeric", length=9)
w_old[1] <- 1000000 * average_jackpot
w_old[2] <- 200000
w_old[3] <-  10000
w_old[4] <-    100
w_old[5] <-    100
w_old[6] <-      7
w_old[7] <-      7
w_old[8] <-      4
w_old[9] <-      3

# expected winnings (loss)
# cost to play is $1
expected_winnings_old_rules <- -1 * (1 - sum(p_old)) + sum(p_old * w_old)

# new rules
# choose 5 from 59 white balls, 1 from 35 red balls
# probabilities of winning powerball
powerball_new <- function(w, r)
    ch(5,w)*ch(54,5-w)/ch(59,5)*ch(1,r)*ch(34,1-r)/ch(35,1)

# 9 ways to win
p_new <- vector("numeric", length=9)
p_new[1] <- powerball_new(5, 1) # five white + powerball_new
p_new[2] <- powerball_new(5, 0) # five white
p_new[3] <- powerball_new(4, 1) # four white + powerball_new
p_new[4] <- powerball_new(4, 0) # four white
p_new[5] <- powerball_new(3, 1) # three white + powerball_new
p_new[6] <- powerball_new(3, 0) # three white
p_new[7] <- powerball_new(2, 1) # two white + powerball_new
p_new[8] <- powerball_new(1, 1) # one white + powerball_new
p_new[9] <- powerball_new(0, 1) # powerball_new

# winnings
# jackpots start $20M larger than before
# second prize is $1M instead of $0.2M
# prize for powerball only is $4 instead of $3
w_new <- vector("numeric", length=9)
w_new[1] <- 1000000 * (average_jackpot + 20)
w_new[2] <- 1000000
w_new[3] <-  10000
w_new[4] <-    100
w_new[5] <-    100
w_new[6] <-      7
w_new[7] <-      7
w_new[8] <-      4
w_new[9] <-      4

# expected winnings (loss)
# cose to play is $2 instead of $1
expected_winnings_new_rules <- -2 * (1 - sum(p_new)) + sum(p_new * w_new)
I found this picture apropos (from http://www.toxel.com/inspiration/2009/09/18/12-creative-toilet-paper-designs/): 

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