  This paper is a comparison of two theories of the probability of a history in
quantum mechanics. One is derived from Copenhagen quantum mechanics using the
projection postulate and is the basis of the "consistent histories"
interpretation; the other is based on a proposal by Bell, originally for the
"pilot state" theory but here applied to pure unitary quantum mechanics. The
first can be used for a wider class of histories but depends on the projection
postulate, or "collapse", which is widely held to be an unsatisfactory feature
of the theory; the second can be used in a theory of the universal state vector
without collapse. We examine a simple model based on Wigner's friend, in which
Bell's model and the projection postulate give different probabilities for the
histories of a sentient system. We also examine the Frauchiger-Renner extension
of this model, in which comparison of the two calculations of histories throws
light on the contradiction found by Frauchiger and Renner. By extending the
model to equip the observer with a memory, we reduce the probability of
histories to the use of the Born rule at a single time, and show that the Born
rule, with the memory, gives the same result as applying projection in the
course of the history, because of entanglement with the memory: entanglement
implements collapse. We discuss the implications of this for the use of
histories in quantum cosmology.
