Tossing a biased coin — Part 2

If John has to select one of three alternatives, he tosses the coin in batches of three until he comes up with a mixed group. Here’s one way he can assign options to coin patterns.

THH or HTT — Option 1
HTH or THT — Option 2
HHT or TTH — Option 3

Each set of patterns has the same probability

p^2(1-p) + p(1-p)^2 = p(1-p)

The same strategy lets him choose between n alternatives, although things do get messy for n > 5.

The binomial calculation helps. The fact that

\binom{5}{2} = 10

means there are 10 ways in which 2 heads and 3 tails can be arranged. So if John has 10 options he needs only batches of five tosses.

Cryptic? Yes. I leave it to you to expand the idea, although it’s drying up.



About aharmlessdrudge

Way back during the late Bronze age -- actually it was the 1950s -- all of us in high school had to take a vocational test to determine our interests and, supposedly, our future careers. I cannot remember the outcome, but I do recall one question that gave me pause. "If you were to win a Nobel prize, would it be in literature or in physics?" I hesitated over the question: although I enjoyed mathematics and science more than English class, I did have a couple of unfinished (and very bad) novels hidden away at home. I cannot remember what I chose back then, but the dilemma followed me to university, where I switched from mathematics to English and -- after a five-year stint in journalism -- back to mathematics. I recently retired as a professor of statistics. Retirement. What a good chance to revive my literary ambitions. I have finished a novel -- more about that in good time -- and a rubble of drafts of articles about mathematics and statistics is taking up space on my hard disk.
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