# Segal's law

Segal's law is an adage that states:

"A man with a watch knows what time it is. A man with two watches is never sure."[1]

The mood of the saying is ironical. While at a surface level it appears to be advocating the simplicity and self-consistency obtained by relying on information from only a single source, the underlying message is to gently question and make fun of such apparent certainty – a man with one watch can't really be sure he knows the right time, he merely has no way to identify error or uncertainty.

Nevertheless, the saying is also used in its purely surface sense, to caution against the potential pitfalls of having too much potentially conflicting information when making a decision.

## In reality

In reality a man possessing one watch has no idea whether it is the correct time unless he is able to compare it to a known standard, in which case he effectively has more than one watch.[2] This situation is not made any worse by having two watches. One might even think that it is better since if the two watches are in approximate agreement one might assume that both are working and an average of them will yield the correct time to within some accuracy depending on the specification of the timepieces. While this is true, the probability of knowing the right time is still exactly the same as with one watch. This is because the probability of all combinations of states of the two watches needs to be taken into account. Let there be two states: W (working—showing the correct time), and B (broken—showing the incorrect time). The set of possible states of the two watches are then;

${\displaystyle S={\text{(WW, WB, BW, BB)}}}$

If the probability of a watch being in the W state is p and in the B state is q, and assuming both watches have the same probability of working, then the total probability of all possible states is;

${\displaystyle p^{2}+pq+qp+q^{2}=p^{2}+2pq+q^{2}=1}$

since it is certain the watches are in one of these states. The first term, p2 represents both watches in the working state so this state will unconditionally yield the correct time. The second term 2pq represents one watch working and the other not. Since it is impossible to know which one is correct one can only guess. Half the time the guess will be right and half wrong so the effective probability of having the right time from this state is only pq. The last term represents both watches not working which will never yield the correct time. The total probability, P, of having the correct time is thus,

${\displaystyle P=p^{2}+pq}$

and since q = 1 − p

${\displaystyle P=p^{2}+p(1-p)=p}$

that is, the same probability as one watch. An improved probability of obtaining the correct time is only possible with at least three watches since majority voting logic can then be applied.[3] The case of three watches has a total probability of,

${\displaystyle p^{3}+3p^{2}q+3pq^{2}+q^{3}=1}$

The second term will always yield the correct time by majority voting. The third term represents two malfunctioning watches. It is possible to tell that there is a problem but not which watch is correct. Thus again, the best solution is a simple guess that will only be right one third of the time. Thus, the total probability of having the correct time is,

{\displaystyle {\begin{aligned}P&=p^{3}+3p^{2}q+pq^{2}\\&=p+p^{2}(1-p)\end{aligned}}}

which is clearly greater than p. Likewise, the probability function of n watches can be found from the binomial expansion of (p + q)n.

This reasoning is not valid if there are systematic errors present in the watches. For instance, if all the watches start to gain at high temperature in the same way this is an error that cannot be either corrected or even detected by majority voting.[2]