Monday, November 4, 2013

Calculating the difficulty of twins (Portraits 42/52)

I photographed two pairs of twin babies few weeks ago.  When I got a booking for twins, I can't help but geeked out and tried to calculate the difficulty of photographing twins numerically, compared to single babies. So here it goes:

In my terms, difficulty depends on how likely the baby goes to sleep.

Probability of a baby goes to sleep = p, where  (p<1 br=""> Difficulty of photographing a single baby = x/p, where x is some multiplier for dependency. x includes all things as feeding, diaper changes and all unhappiness of the babies.

Probability of two babies go to sleep at the same time = P{baby A goes to sleep} * P{baby B goes to sleep} = p*p = p^2 ,
1) assuming P{baby A..} = P{baby B ...} = p  (of course this is too generic to say all babies are likely to sleep with same probability, but for simplicity) and
2) assuming two babies are indepedent variables in order to simplify conditional probabilities, meaning one sleeping doesn't make the other one more/less likely to sleep.

Then twins' difficulty = x/p^2, assuming multiplier x is the same for both singles and twins.  (using another y would be more appropriate)
Therefore twins are more difficult than singles by (x/p^2) / (x/p)  = 1/p.

e.g: if p = 1/2
P{two babies go to sleep at the same time} = 1/4
Twins are more  difficult than singles by 2 times.

With triplets, they are harder than singles by 1/p^2 = 4 times

Of course, this super-simplified analysis considers sleep as the only difference between twins and singles. Otherwise, I need to introduce a lot more variables. With this difference alone, twins are twice harder than singles.

Professor Sahai at Berekley would be either proud of me that I'm applying myself or so ashamed of me that I didn't do this right (more likely) or ashamed that I'm making this easy problem into a huge calculation (the most likely). Or like my dad always says: "those of you born in calculator age can't do quick math in your heads". So, for those of us (me) born in statistical modeling age can't do probability math in my head.