## Random Error Theory

Probability - The ratio of the number of times that an event should occur to the total number of possibilities - If event A can occur m ways and fails to occur n ways, then the probability of event A occurring is m/(m + n) - Probability of throwing a 1 on a fair die - Should occur one time - Will fail to occur 5 times - Probability is 1/(1+5) = 1/6 - Probability of an event failing to occur is 1 − m/(m + n) |

- Probability is always between 0 and 1

- 0 means there is no chance of occurrence

- 1 means that the event will absolutely occur

- 0 means there is no chance of occurrence

- 1 means that the event will absolutely occur

**Compound Events**

- Compound event: The simultaneous occurrence of two or more events.

- The probability of a compound event is the product of each probabilities of each individual event occurring. P = P1 x P2 x … x Pn.

**Another Example**

- Assume that a tape exists that can only make a +1 ft or −1 ft when taping a distance.

- Let t be the number of ways that each error can occur

- And T be the total number of possibilities

- Then the possible occurrence of these random errors is...

- For a distance of 1 tape length

- Only one +1 error or one −1 can occur

- For a distance of 2 tape lengths

- +1 and +1 occurs once with a resulting combined error (ce) of +2

- +1 and −1 can occur twice (ce = 0)

- −1 and −1 occurs once (ce = −2)

- For a distance of 3 tape lengths

- +1, +1, +1 error occurs once (ce = +3)

- −1, +1, +1; +1, −1, +1; +1, +1, −1 (ce = +1) occurs 3 times

- −1, −1, +1; −1, +1, −1; +1, −1, −1, (ce = −1) occurs 3 times

- −1, −1, −1; (ce = −3) occurs 1

- And so on…

- Now

- Draw a single line across the target tabulating the number of BBs that pierced the target along the line

- Create a frequency plot of the hits versus distance from the center of the target

- It may look as follows

- Draw a single line across the target tabulating the number of BBs that pierced the target along the line

- Create a frequency plot of the hits versus distance from the center of the target

- It may look as follows