Re: [math-fun] Probability puzzle
Okay, now suppose the same question but suppose for given p+q = 1, a large number of people flip their own biased coin, with Prob(H) = p and Prob(T) = q = 1-p, until reaching the first T. What will be the asymptotic proportion of H's and T's flipped, in simplest form? --Dan P.S. I had to solve this problem before I appreciated the reasoning behind Gene's beautiful solution, which didn't register with me at first. << PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population? (Of course, make all the usual simplifying assumptions in this kind of problem.)
I sleep as fast as possible so I can get more rest in the same amount of time.
Yes. I suppose a simple rephrasing of the problem... Suppose you continually flip a coin, with p and q as you describe. Notice every time you get H. What is the fraction of Ts? Another question is, are there plausible rules of heredity and reproduction that would alter the Boy/Girl ratio if "families" ceased reproducing at the first boy, but not if they stopped randomly? --ms On Wednesday 07 July 2010 23:06:05 Dan Asimov wrote:
Okay, now suppose the same question but suppose for given p+q = 1, a large number of people flip their own biased coin, with
Prob(H) = p and Prob(T) = q = 1-p,
until reaching the first T.
What will be the asymptotic proportion of H's and T's flipped, in simplest form?
--Dan
P.S. I had to solve this problem before I appreciated the reasoning behind Gene's beautiful solution, which didn't register with me at first.
<< PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
I sleep as fast as possible so I can get more rest in the same amount of time.
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On Thu, Jul 8, 2010 at 8:08 AM, Mike Speciner <ms@alum.mit.edu> wrote:
Another question is, are there plausible rules of heredity and reproduction that would alter the Boy/Girl ratio if "families" ceased reproducing at the first boy, but not if they stopped randomly?
Suppose different couples have different chances of having a boy. For example, half the couples have a 75% chance that each of their children is a boy, while half the couples have a 25% chance that each of their children is a boy. Now a "stop at the first boy" rule will change the ratio in the population. Andy
OK, but what are the plausible heredity rules that preserve the relevant ratios in the steady state (i.e., without the termination rule)? (By "plausible", I mean conforming to standard human genome kinds of things--gender determination from XvY sperm, mitochondrial genome from mother, mother-father gene pairs for everything else.) Presumably, the children of the two kinds of couples are genetically different in some way. What are the coupling rules that preserve the ratios, remembering that you need a boy to couple with a girl, and we probably need to assume monogamy to apply the "stop at first boy" rule (although it could be applied to just one parent, taking just that parent out of the reproducing pool). Assume a finite population, so at least some "interracial" coupling is required in the steady state. --ms On Thursday 08 July 2010 10:00:13 Andy Latto wrote:
On Thu, Jul 8, 2010 at 8:08 AM, Mike Speciner <ms@alum.mit.edu> wrote:
Another question is, are there plausible rules of heredity and reproduction that would alter the Boy/Girl ratio if "families" ceased reproducing at the first boy, but not if they stopped randomly?
Suppose different couples have different chances of having a boy. For example, half the couples have a 75% chance that each of their children is a boy, while half the couples have a 25% chance that each of their children is a boy. Now a "stop at the first boy" rule will change the ratio in the population.
Andy
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Mike Speciner