Should I join a group, or stay an individual?

Question

I have just taken part in a draw and it has left me wondering whether I made the right decision to enter the draw as an individual or whether I should have entered as a group, and if so, what size group would have been optimal.

For this draw my partner and I signed up as individuals. Neither of us wanted to deprive the other of a place if we didn't get in, but since I got a place and she didn't, and I now know the mechanics of the draw, I'm prompted to question our choices.

The set up is as follows.

• A game has 30 places for male players and 30 places for female players.^†
• 100 players have signed up for the game (half male and half female^†), so not everyone can play.
• Players can either register as individuals, or register in a group, such as a couple, family or fraternity of friends. No-one can register more than once.
• Individuals are assigned a single draw number. If an individual is selected before all places for their gender are filled they get in the game, otherwise they are placed on a wait list.
• Groups are also assigned a single draw number. If a group is selected before all places required for that group are taken they all get in the game, otherwise they are all placed on a wait list.

For example, let's say the distribution is as follows:

• 35 Male
• 35 Female
• 11 couples (FM)
• 2 fraternities of four (MFFM)

This means that there are 83 draw numbers, but each draw could allocate 1, two or 4 places, or add that number of players to the wait list.

As an individual, do I have the same chance of getting a place as I would if I were registered as a couple? What about if I registered as part of a larger group?

Looking at this naively, I would assume that since there are 30 places for 50 players of each gender, both my partner and I would have a 60% chance of getting a place, a 36% chance that both of us would get a place and a 16% chance that neither of us would get a place.

Even more naively I assumed that if we registered as a couple, we would still have a 60% chance of getting a place, and thus 40% chance of not getting a place, but is that true?

^{† This is a simplification, the actual draw also had a small number of gender neutral players and places.}

Answer 1

You personally have a higher chance of getting a place if you enter as an individual. To see this, imagine that the other entries are fixed, and you either enter two separate individuals, or a couple and a blank piece of paper in the draw (this is to keep the number of things in the draw the same, but if the blank paper gets drawn out it just gets ignored). Now we order the things in the draw, and condition on knowing what order everything will be drawn except which of those two is which (so you know the positions of the two pieces of paper relevant to you, but not which is which, and you know the position of everything else). Now given this information, and whether you entered as a couple or an individual, you can tell whether you get a place in $0$, $1$ or $2$ of the $2$ possible arrangements. But this number will always be at least as large if you enter as an individual as it is if you enter as a couple, since if when the first (say) piece of paper is drawn there is enough room for a couple, there is certainly enough room for you as an individual.

edit: The only difficult case above is as follows (thanks to ingix for pointing this out). Is it possible that one of you (say, male) being chosen first prevents the other from being chosen second, but owing to a different subset of the groups drawn in between being given places, you would have gotten through as a couple in either position? Actually it isn't. Suppose that from the first paper one of you gets a place as an individual male. In order for this to change which groups get through between the two papers, there must be a group which would have gotten through had the first paper been blank, but didn't because you took an extra male spot. In that case, if that group did get through it would use up the last male slot, so you wouldn't get through as a couple.

This gives you at least as high a probability of getting a place as an individual. Provided there are some people entering as individuals, there are some arrangements of the others where you do strictly better as an individual.

For the example entries you suggest, I wrote a python program to simulate the draw. It suggests that the individuals have roughly a $61.5\%$ chance, the couples $56.8\%$ and the larger groups $55.9\%$ (based on a million runs).

from random import shuffle

draw=[(0,1)]*35+[(1,0)]*35+[(1,1)]*11+[(2,2)]*2
success={(0,1):0,(1,0):0,(1,1):0,(2,2):0}
runs=1000000
for i in range(runs):
    shuffle(draw)
    m,f=0,0
    for x in draw:
        if x[0]+m<=30 and x[1]+f<=30:
            m+=x[0]
            f+=x[1]
            success[x]+=1
for x in success:
    print("{}\t{:.1f}".format(x,100*success[x]/(draw.count(x)*runs)))