Random Error Theory

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

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 …

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

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

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 = 0.3497

Normal Distribution

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

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

Normal Distribution Function

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.

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

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