Saturday, January 5, 2013

Matt's probability question

A quick post since I don't have access right now to my beautiful plotting software.

My friend Matt asks a question: if I have 12 cards labelled 1-12, and each day I draw 3 without replacement, record which ones I've drawn, and then shuffle them back in for the next day, how many days should I expect to draw cards before I have drawn all 12?

Instead of doing actual math, I just used MATLAB and ran one million simulations (a nice round number). First I'll plot the probability of having drawn all 12 cards by day X

You can see that we cross 50% somewhere between day 11 and 12. Specifically, 46.4% of simulations were done by day 11, and 57.1% were done by day 12; Interpolating gives about 11.3. Even though I'm too lazy to do stat math for you, we can see the form of the equation if we plot the probability to not be done yet versus number of days (below). In a semilog (y) plot, it forms a straight line after about day 10.

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