Here we are concerned with the case in which an event must
be one of two classes, such as up or down, forward or back,
positive or negative, etc. Let *p* be the probability for an
event of Class 1. Then (1 - *p*) is the probability for Class 2,
and the joint probability for observing *N*_{1} events in
Class 1 out of *N* total events is

The binomial distribution | (14) |

Note that
_{j=1}^{N}
*p*(*j*, *N*) = [*p* + (1 - *p*)]^{N} =
1. The factorials correct
for the fact that we are not interested in the order in which
the events occurred. For a given experimental result of
*N*_{1} out of *N* events in Class 1, the likelihood
function
(*p*) is then

(15) | |

(16) |

From Eq. (15) we have

(17) |

From (16) and (17):

(18) |

The results, Eqs. (17) and (18), also happen to be the same as those using direct probability. Then

and

__Example 4__

In Example 1 on the *µ*-e decay angular distribution we found
that

is the error on the asymmetry parameter
. Suppose that the
individual cosine, *x*_{i}, of each event is not known. In this
problem all we know is the number up vs. the number down. What
then is
? Let *p* be the probability
of a decay in the up hemisphere; then we have

By Eq. (18),

For small this is
=
sqrt[4 / *N*] as compared to
sqrt[3 / *N*] when the full information is used.