Suppose it is known that either Hypothesis A or Hypothesis
B must be true. And it is also known that if A is true the
experimental distribution of the variable *x* must be
*f*_{A}(*x*), and
if B is true the distribution is *f*_{B}(*x*). For
example, if
Hypothesis A is that the K meson has spin zero, and hypothesis
B that it has spin 1, then it is "known" that
*f*_{A}(*x*) = 1 and
*f*_{B}(*x*) = 2*x*, where *x* is the kinetic
energy of the decay ^{-}
divided by its maximum value for the decay mode
*K*^{+} ->
^{-} +
2^{+}.

If A is true, then the joint probability for getting a
particular result of N events of values
*x*_{1}, *x*_{2},..., *x*_{N} is

The likelihood ratio is

(1) |

This is the probability, that the particular experimental result
of *N* events turns out the way it did, assuming A is true, divided
by the probability that the experiment turns out the way it did,
assuming B is true. The foregoing lengthy sentence is a correct
statement using direct probability. Physicists have a shorter
way of saying it by using inverse probability. They say
Eq. (1) is the betting odds of A against B. The formalism of
inverse probability assigns inverse probabilities whose ratio
is the likelihood ratio in the case in which there exist
no prior probabilities favoring A or B.
[3] All the remaining
material in this report is based on this basic principle alone.
The modifications applied when prior knowledge exists are
discussed in Sec. 10.

An important job of a physicist planning new experiments
is to estimate beforehand how many events he will need to
"prove" a hypothesis. Suppose that for the *K*^{+} ->
^{-} +
2^{+} one
wishes to establish betting odds of 10^{4} to 1 against spin 1.
How many events will be needed for this? The problem and the
general procedure involved are discussed in
Appendix I: Prediction of Likelihood Ratios.