Theory of Probability
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Probability
The ratio of the number of ways an event can occur to the total number of possible outcomes.
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Event
An outcome or group of outcomes of an experiment.
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Compound Event
An event involving the simultaneous occurrence of two or more events.
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Independent Events
Events in which the occurrence of one does not affect the probability of another.
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Random Error
An unpredictable measurement error that may be positive or negative.
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Normal Distribution
A bell-shaped probability distribution commonly used to describe random measurement errors.
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Probable Error
A range around the mean within which a specified percentage of observations is expected to fall.
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P(E) = Number of favorable outcomes / Total number of possible outcomes
Basic Probability Formula:
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• Probability of occurrence: P(E) = m / (m + n)
• Probability of failure: P(not E) = n / (m + n)
• Probability of failure: P(not E) = n / (m + n)
If an event can occur in m ways and fail to occur in n ways:
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0 to 1
Probability ranges from
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P(E) = 0
means the event is impossible.
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P(E) = 1
means the event is certain.
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1
The sum of all possible outcome probabilities equals
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Compound Event
The simultaneous occurrence of two or more events.
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P = P₁ × P₂ × P₃ × ... × Pₙ
Where:
• P — Probability of simultaneous occurrence.
• P₁, P₂, P₃, ... — Individual event probabilities.
• n — Number of independent events.
Where:
• P — Probability of simultaneous occurrence.
• P₁, P₂, P₃, ... — Individual event probabilities.
• n — Number of independent events.
Probability of Compound Events
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• +1 — Positive error.
• −1 — Negative error.
• −1 — Negative error.
Since the error is random, there are two possible values:
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T = 2ⁿ
Where:
• T — Total number of possible combinations.
• n — Number of combined measurements.
Where:
• T — Total number of possible combinations.
• n — Number of combined measurements.
Total Number of Possibilities:
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distribution of random errors;
bell-shaped
bell-shaped
As the number of combined measurements increases, the ___ approaches a normal distribution, which has a ___ curve.
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Normal Distribution Curve
A bell-shaped curve representing the distribution of random errors.
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Probability Density Function (PDF)
A mathematical function describing the distribution of probabilities.
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Standard Deviation (σ)
Measures the spread of observations around the mean.
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Mean (μ)
The central or average value of a distribution.
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f(x) = [1 / (σ × √(2π))] × e^[−(x − μ)² / (2σ²)]
Where:
• f(x) — Probability density function.
• e — Base of natural logarithms (approximately 2.71828).
• x — Value of the random variable or error.
• μ — Population mean.
• σ — Population standard deviation.
• π — Approximately 3.14159.
Where:
• f(x) — Probability density function.
• e — Base of natural logarithms (approximately 2.71828).
• x — Value of the random variable or error.
• μ — Population mean.
• σ — Population standard deviation.
• π — Approximately 3.14159.
Normal Probability Density Function:
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f(x) = [1 / (σ × √(2π))] × e^[−x² / (2σ²)]
For a Normal Distribution with Mean Zero:
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f(μ) = 1 / (σ × √(2π))
Central Ordinate:
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• Mean (μ) = 0
• Variance (σ²) = 1
• Standard deviation (σ) = 1
• Variance (σ²) = 1
• Standard deviation (σ) = 1
For a standard normal distribution:
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Standard Normal Distribution
A normal distribution with a mean of 0 and a standard deviation of 1.
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Z-Value (Standard Score)
Indicates how many standard deviations a value is from the mean.
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Standard Normal Distribution Table
A table used to determine the probability or area under the normal curve.
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z = (x − μ) / σ
Where:
• z — Standard score.
• x — Observed value.
• μ — Mean.
• σ — Standard deviation.
Where:
• z — Standard score.
• x — Observed value.
• μ — Mean.
• σ — Standard deviation.
Formula for the Z-Value:
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z = (x − μ) / σ
Converting a Normal Distribution into a Standard Normal Distribution:
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P(Z ≤ z) = Φ(z)
• Φ(z) — Area under the standard normal curve from negative infinity up to z.
• Φ(z) — Area under the standard normal curve from negative infinity up to z.
Standard Normal Cumulative Probability:
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standard normal distribution table
provides cumulative probabilities.
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P(Z > z) = 1 − Φ(z)
To Find the Probability Above a Z-Value:
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P(a < Z < b) = Φ(b) − Φ(a)
Where:
• a — Lower z-value.
• b — Upper z-value.
• Φ(b) — Cumulative probability at the upper limit.
• Φ(a) — Cumulative probability at the lower limit.
Where:
• a — Lower z-value.
• b — Upper z-value.
• Φ(b) — Cumulative probability at the upper limit.
• Φ(a) — Cumulative probability at the lower limit.
Probability Between Two Values
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normal distribution
is symmetrical around its mean.
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Φ(−z) = 1 − Φ(z)
Symmetry of the Normal Distribution
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P(−z < Z < +z) = 2Φ(z) − 1
Probability Between Equal Positive and Negative Z-Values:
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68.3%
For a normal distribution, approximately ___ of observations fall within one standard deviation above or below the mean.
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±1σ
Approximately 68.3% of observations fall within this range.
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±2σ
Approximately 95.45% of observations fall within this range.
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±3σ
Approximately 99.73% of observations fall within this range.
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Percent Probable Error
The error range around the mean corresponding to a specified probability percentage.
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Eₚ = k × S
Where:
• Eₚ — Probable error at a specified percentage.
• k — Multiplier corresponding to the desired probability.
• S — Standard deviation of the observations.
Where:
• Eₚ — Probable error at a specified percentage.
• k — Multiplier corresponding to the desired probability.
• S — Standard deviation of the observations.
Percent Probable Error
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x̄ = Σx / n
Arithmetic Mean
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S = √[Σ(x − x̄)² / (n − 1)]
Sample Standard Deviation
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E₅₀ = 0.6745S
50% Probable Error
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E₉₅ = 1.960S
95% Probable Error
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• 50% range: x̄ ± E₅₀
• 95% range: x̄ ± E₉₅
• 95% range: x̄ ± E₉₅
Acceptable Ranges
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Percentage = (Number of observations within the range / Total observations) × 100
Percentage of Observations Within a Range
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