# Celestial navigation explained - page 8

## Random errors and common sense

Random errors are inevitable... We can find in the Bowditch chapter 16 (Nathaniel Bowditch, __The American
Practical Navigator, an Epitome of Navigation__, pub. n°9 NIMA, USA, 1995) 19
possible errors when observing the celestial body height and 30 possible errors until the LOP
can be drawn on the chart.

In case of random errors, without further statistical analysis, the True Position can be
on the left or on the right of each LOP, with an equal probability.

Each LOP divides the world in 2 areas and the True Position has exactly 50% of
chance to be in one of them.

Let's call (arbitrarily) the zone inside the triangle +
+ +.
The names of the other zones follow directly:

There is no - - - zone. The True
Position just cannot be in the **- area** of each LOP and the cocked hat
still be shaped as drawn!

__So we know that the True Position is in the + area of at least 1
position line__.

The True Position can be found by flipping a coin twice (head side is +, reverse side is - and each occurrence has a 50% probability).

1) If the True Position is in the + +
+ zone, it needs to be in 3 **+ areas**:

- we know that we are in 1
**+ area** - flip the coin, there is 1 chance among 2 to be in a second
**+ area** - flip the coin again, there is also 1 chance among 2 to be in a third
**+ area** - so there is 1/2 x 1
**/**2 = 1/4 chance to be in the +++ zone = 25% probability

2) If the True Position is in a - +
+ zone, it needs to be in 2 **+ areas** and 1 **-
area**:

- we know that we are in 1
**+ area** - flip the coin, we are in 1
**+ area**or 1**- area** - flip the coin again, there is now 1 chance among 2 to be in the remaining
**+**or**- area** - so there is 1/2 chance (50%) to be in a -++ zone
- as there are 3 -++ zones, 50% divided by 3 = 16.67% probability

3) If the True Position is in the -- +
zone, it needs to be in 1 **+ area** and 2 **- areas**:

- we know that we are in 1
**+ area** - flip the coin, there is 1 chance among 2 to be in a -
**area** - flip the coin again, there is also 1 chance among 2 to be in a second
**- area** - so there is 1/2 x 1
**/**2 = 1/4 chance (25%) to be in a --+ zone - as there are 3 --+ zones, 25% divided by 3 = 8.33% probability

So by 1, 2 and 3, we can draw:

### Conclusion

The True Position is the cocked hat centre if there are no random errors and the observations azimuths are spread over more than 180°. If there are random errors, then the most probable position (MPP) is inside
the cocked hat but with a probability of only 25%. |