1. Random Error Theory

2. Introduction
From now on, we will assume that systematic errors have been eliminated
This leaves random errors and possible blunders
Random errors conform to the laws of probability

3. Probability
Defined as the ratio of the number of times an event should occur to the total number of possibilities
For example, tossing a “2” with a fair die will have a probability of 1/6
When an event can occur in m ways and fail to occur in n ways …

4. Probability - continued
Probability ranges from 0 to 1
Sum of all probabilities for an event must be one
Compound events are two independent events that occur simultaneously
Their probability is the product of the independent probabilities of the events
The probability of tossing two 2’s in a row is (1/6)x(1/6) = 1/36

5. What is probability that 2 red balls will be drawn when one ball is randomly selected from each box? Box 1: 1W, 2R, 3W, 4W; Box 2: 1R, 2R, 3W, 4W, 5W
TR BOX 1 BOX 2
1 1 WHITE1 1 RED1
2 1 RED2 1 RED1
3 1 WHITE3 1 RED1
4 1 WHITE4 1 RED1
5 1 WHITE1 1 RED2
6 1 RED2 1 RED2
7 1 WHITE3 1 RED2
8 1 WHITE4 1 RED2
9 1 WHITE1 1 WHITE3
10 1 RED2 1 WHITE3
P( 2 REDS ) = 2/20 = 1/10 = 1/4 × 2/5 = 2 /20 = 1/10
TR BOX 1 BOX 2
11 1 WHITE3 1 WHITE3
12 1 WHITE4 1 WHITE3
13 1 WHITE1 1 WHITE4
14 1 RED2 1 WHITE4
15 1 WHITE3 1 WHITE4
16 1 WHITE4 1 WHITE4
17 1 WHITE1 1 WHITE5
18 1 RED2 1 WHITE5
19 1 WHITE3 1 WHITE5
20 1 WHITE4 1 WHITE5

6. The following 5 cards are showing from a standard deck of cards
The following 5 cards are showing from a standard deck of cards. What is the probability that one or both of the next two cards are clubs (complete the flush)?
Ace of clubs, 6 of clubs, 5 of spades, 10 of clubs, Jack of clubs
Winning possibilities:
Card 1 – club, card 2 – not club
Card 1 – not club, card 2 club
Card 1 – club, card 2 club
Total =

7. Normal Distribution

8. Normal Distribution
As n approaches infinity, the histograms will approach the normal distribution (AKA bell) curve

10. Properties of the Normal Distribution
Area under the curve from -∞ to +∞ = 1
Slope = 0 at x
Inflection points at ±σ from x

12. Normal Distribution Function

13. Standard Normal Distribution Function
For a mean of zero (μ=0) and standard deviation of one (σ=1)
See NORMDIST and NORMSINV spreadsheet functions or STATS program.

14. Various Errors Standard Error (±σ) = 68.3% E50 = 0.6745 σ
(look at STATS and Excel functions)

16. Example 3.2
50 seconds-readings are given. Find mean, standard deviation, and E95.