Western States Lottery - History and Projections

I've been watching the lottery at Western States now since I started entering it in 2012. 2015 was my lucky year but as I've been tracking the data I thought I'd go ahead and punch in the 2016 lottery entrants data into my tracking spreadsheet and update the progression. After seeing an estimated average 6 year journey starting this year with one ticket for me to get in again and knowing I wasn't willing to put in that journey for a repeat (I had a glorious 2015 WS - and got my Silver Buckle) - I elected to withdraw my name from the hat and help the odds of everyone else ever so slightly :). Good Luck for those trying to get in. As I'm moving on now most likely I won't be watching this into the future - or maybe I will just out of curiosity how my predictions turned out.

History:

2000 – 583 applicants.
2005 – 791 applicants.
2008 – 1,350 applicants. (year of the fire cancellation)
2010 – 1,695 applicants.
2011 – 1,786 applicants. (first year of the multiple tickets solution)
2012 – 1,940 applicants
2013 – 2,295 applicants (trimmed down qualifier race list to larger ~130 races)
2014 - 2705 applicants
2015 - 2566 applicants (Cut out 50 mile qualifiers, added 2^n-1 Jan, 2015)
2016 - 3526 applicants (74 qualifier races)

......Enter...1(1)...2(2)..4(3)..8(4)..16(5)..32(6)..64(7)..Tickets Odds(1) # for >50%
2011..1,786...1,286..500....................................2,286...10%....7
2012..1,940...1,221..461...258..............................2,917...9.4%....7
2013..2,295...1,486..480...207...122........................3,555...8.0%....9
2014..2,705...1,727..561...258...106...53...................4,312...6.5%...11
2015..2,566...1,427..641...281...136...57....24.............6,601...4.7%...15 (Last for 5 yr >50%)
2016..3,526...2,269..623...372...170...72....15......5......8,315...3.6%...19

* Odds for 1 approximated based on X winners (assume 2016 is X=270 as was 2013-2015) and a normalized win rate across the tickets. To account for multiple tickets instead of straight odds X/Total Tickets in the lottery - the denominator is adjusted down to refect the average active tickets by predicting the number of inactive tickets in the hat at the end of the lottery and taking 1/2 of those out of the denominator so for example 2016 the inactive tickets at the end is estimated at 1720 so the odds are 270/[8315-1720/2)]=3.62% - compared to the more detailed montecarlo run of 3.68 - approximation is pretty close. 

Projections for the future must require assumptions - these are the ones used below:

- 46% of those with 1 ticket that don't win will requalify/enter lottery next year - same rate as 2015 into 2016 lottery. 63% with 2 tickets, 73% with 4 tickets, 80% with 8, 75% with 16, 83% with 32 or more - all same rate as 2015 into 2016. Assume these rates continue into future years.
- 20% growth rate in those with 1 tickets - last years growth from 1427 to 2269 (59%) of those entering the hopper I'm assuming is a one time fluke perhaps and will normalize back toward more historical levels or other factors such as available qualifying races will keep the growth rate lower.
- Entry rules going forward stay the same.

Here's the projected counts

 ......Enter...1(1)...2(2)..4(3)..8(4)..16(5)..32(6)..64(7)..128(8).Tickets Odds(1) # for >50%
2016..3,526...2,269...623..372...170....72.....15.....5.....0......8.3k....3.6%....19
2017..4,464...2,723..1001..368...236...102.....30.....4.....0......11k.....2.7%....25 (last for 6>50%)
2018..5,537...3,267..1212..602...242...152.....50....11.....1......15k.....2.0%....34
2019..6,806...3,921..1465..740...407...165.....83....22.....2......20k.....1.5%....47 (last for 7>50%)
2020..8,326...4,705..1767..903...511...290.....98....43.....7......27.5k...1.1%....65

 Someone with one ticket in 2016 lottery would expect the odds progression:
.....Year Total

2016 3.6% 3.6%
2017 5.3% 8.6%
2018 7.7% 15.8%
2019 11.1% 25.2%
2020 15.8% 37.0%
2021 22.0% 50.9%
2022 30.2% 65.7%

Total column is the cumulative odds so by 2021 there's a 50.9% chance of being picked at some lottery between 2016 and 2021. 

Two tickets in 2016 progression:

.....Year Total

2016 7.1% 7.1%
2017 10.4% 16.8%
2018 14.9% 29.2%
2019 21.0% 44.1%
2020 29.0% 60.3%
2021 39.2% 75.9%
2022 51.3% 88.3%

Four(3 yrs) tickets in 2016 progression:

.....Year Total

2016 13.7% 13.7%
2017 19.7% 30.7%
2018 27.6% 49.8%
2019 37.7% 68.7%
2020 49.7% 84.3%
2021 63.1% 94.2%
2022 76.3% 98.6%

Eight (4 yrs) tickets in 2016 progression:

.....Year Total
2016 25.6% 25.6%
2017 35.5% 52.0%
2018 47.5% 74.8%
2019 61.1% 90.2%
2020 74.6% 97.5%
2021 86.4% 99.7%
2022 94.4% 100.0%

The growthrate of 1 ticket holders is a very sensitive variable. Assumption above is the growth of the 1 ticket holders averages 20% even though the 2015 to 2016 growth was 59% which is probably optimistic. At that rate in 2020 a 7 year streaker with 64 tickets would have about 50% odds to get in (65 tickets as shows above is actually the >50% number of tickets). Here's a few sensitivities:

20% growth in 1 ticket holders each year = 65 tickets for >50% in 2020
30% = 76 tickets for >50% odds in 2020.
40% = 89 tickets
50% = 104 tickets
60% = 121 tickets

No idea what will actually happen - demand has exceeded expectations every year I've tried to predict the next year. This year's 59% demand growth in first timers and 37% growth in total entrants was way bigger than I would have expected. The number of races on the qualifying list does appears to be trending up for 2017 (83 vs 74 in 2016) to support the increased demand so with that trend continuing and the continued draw to the sport in general - I suppose the trend could keep going - not really sure what rule change WS could do next to curb the demand short of making registration $10k :).

Cheers.