SECTION 6.1 Discrete Random Variables 343
(c) That is, ifAstarts witha dollars andbstarts withbdollars,
then the probability thatAbankruptsBisa
a+b.
The point of the above is that ifAplays against someone with a lot
of capital—like a casino—then the probability thatAeventually
goes bankrupt is very close to zero, even if the game is fair! This
is known asgambler’s ruin.- Generalize the results of Exercise 12 to the case when the proba-
bility of tossing head isp. That is, compute the probability thatA
bankruptsB, given thatAstarts withadollars andBhasN−a
dollars. - (An open-ended question) Note that the Poisson distribution with
mean 2 and the geometric distribution withp=.5 both have the
same mean and variance. How do these distributions compare to
each other? Try drawing histograms of both. Note that the same
can be said for the Poisson distribution with mean 2k and the
negative binomial (p=.5, stopping at thek-th success). - Suppose we have a large urn containing 350 white balls and 650
blue balls. We select (without replacement) 20 balls from this
urn. What is the probability that exactly 5 are white? Does this
experiment differ significantly from an appropriately-chosen model
based on the binomial distribution? What would the appropriate
binomial approximation be? - Suppose that we have a large urn containing 1000 balls, exactly
50 of which are white (the rest are blue). Select 20 balls. Without
knowing whether the selection was with or without replacement,
estimate
(a) the expected number of white balls in the sample;
(b) the probability that you selected at most 2 white balls (using
a Poisson model);
(c) the probability that you selected at most 2 white balls (using
a hypergeometric model);