The Horizon casino will hold a tournament in June. For 20 weeks, they will hold a weekly drawing, choosing 6 people each week to give them an entrance to the tournament. After each drawing, all entries are discarded before new ones are accepted for the next week. If you have 20 entries, is it better to put in 1 ticket per week, or put them all in at once, or some other option, assuming all weeks have 7500 entries (other than yours)?
There are busy and slow weeks. A slow week will have less total entries for the drawing. Assuming you can predict a busy/slow week, does the answer change? For numerical purposes, let a busy week have 10000 total entries and a slow week have 5000 before you add your entries.
Source: Real-life experience.
Putting in 1 ticket per week you probability if
winning each week is 6/7501 and your probability of losing each week is
7495/7501 so you probability of losing for each of 20 weeks is
(7495/7501)^20 and your probability of winning at least once is
1-(7495/7501)^20 = .0158768808
Putting in 20 tickets for one week your probability of losing is
7500/7520*7499/7510*...*7495/7515= (7500 C 6)/(7520 C 6) and your
probability of winning at least once is 1 minus this = .0158569596 which is
slightly lower.
With 10 weeks of 5000 other tickets and 10 weeks of 10000 tickets there are
3 choices:
one in each week,
all in one of the 5000 weeks,
two in each of the 5000,
Choice 1 probability = 1-(9995/10001)^10*(4995/5001)^10 = .0178448643 (I
thought this seemed too high until I considered weeks with 10 and 14990
respectively)
Choice 2 probability = 1-(5000 C 6)(5020 C 6) =.0236792298
Choice 3 probability = 1-((4996/5002*4995/5001)^10) = .0237213246
So Choice 3 is the best (it is something of a hybrid of two choices in the
first case.)