Abstract
A classical probability question asks for the expected waiting time
for flipping a coin (fair or not) until a series of consecutive k heads
occur. Now instead of k heads, we can ask for the expected waiting
time for a prescribed string such as HTHHTT (H for ‘head’ and T for
‘tail’), and furthermore, the following more general setting: replacing
coin flipping by taking a letter, one at a time, what is the expected
waiting time until a prescribed string (a series of letters) is reached?
Here we allow different probabilities for the occurrence of different let-
ters. We give an exposition to this problem by offering an elementary
algorithm and implementing it to compute the corresponding probabil-
ity generating function: we show that there exists a universal program
taking as inputs the choice of letters with given probabilities and the
prescribed string, and as output, returning the probability generating
function for the waiting time. The same method is applied to solve the
problem of several competing strings, which asks for the probability (or
more generally the probability generating function) of one of the given
strings occurring before the remaining strings. In particular, this solves
the problem of finding the expectation and variance for the waiting time
random variable of the first problem.
AbstractA classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different let- ters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probabil- ity generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.
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