GEA1000 Chap.4 Probability and Sampling Distributions
Probability and Sampling Distributions
Conditions redefine the sample space
Conditional probability restricts attention to cases satisfying the known condition. Reversing the order usually changes the denominator, which is why sensitivity and the probability of disease after a positive result are different quantities.
Mutually exclusive events and independent events also describe different relationships.
A tree or two-way table turns rates into counts along explicit branches. This is especially useful when the base rate is small, because a modest false-positive rate applied to a large no-condition group can create more positive results than the true-positive branch.
Once the condition is known, total only the branches that satisfy it.
Intersections, unions and partitions
The intersection contains outcomes shared by two events. A union contains outcomes in either event and subtracts the intersection once to avoid double-counting. A partition divides the sample space into non-overlapping branches whose weighted conditional probabilities can be added.
The formulas are bookkeeping rules for the same sets shown in a diagram.
Exclusion differs from independence
Mutually exclusive events cannot occur together. Independent events can occur together, but learning one occurred does not change the probability of the other. Non-empty mutually exclusive events therefore cannot be independent, because one event makes the other impossible.
Test independence through a conditional equality or the product rule rather than through everyday wording.
Repeated samples create a distribution
A sample statistic varies because chance selects different observations. Its sampling distribution is centred on the population parameter under an unbiased design, while standard error describes its repeated-sample spread.
Larger samples reduce this spread through a square-root rule.
Standard error belongs to the statistic, not to individual observations. Quadrupling sample size halves the standard error, while multiplying sample size by nine divides it by three. Larger samples reduce random variability but do not remove coverage, measurement or nonresponse bias.
Precision and validity remain separate judgements.
The Central Limit Theorem sets up inference
For a sufficiently large random sample, the sampling distribution of the mean is approximately normal even when the population is not. Strong skew and heavy tails can require more data.
The theorem does not say the raw observations become normal; it describes the repeated distribution of sample means used by later interval and test procedures.
What this chapter covers
- 01
Events and sample spaces
- 02
Conditional probability
- 03
Mutually exclusive events
- 04
Independent events
- 05
Sampling distributions
- 06
Standard error and the Central Limit Theorem
Worked example · free
Reverse a diagnostic conditional
- 3Translate all rates into counts for a common population.
- 3Condition on positive results and interpret the fraction.
Key terms
- Sample Space
- The collection of possible outcomes relevant to a probability experiment.
- Conditional Probability
- An event probability calculated after restricting the sample space by known information.
- Mutually Exclusive
- A relationship between events that cannot occur together.
- Independent Events
- Events for which learning one occurred does not change the probability of the other.
- Sampling Distribution
- The distribution of a statistic over repeated random samples of one fixed size.
- Standard Error
- The standard deviation of a statistic across its sampling distribution.
Probability and Sampling Distributions FAQ
Why are reversed conditional probabilities different?
They share the intersection in the numerator but condition on different groups. Disease given positive uses all positive results as its denominator, while positive given disease uses everyone with disease. A probability tree or count table keeps both restricted totals visible.
Are mutually exclusive events independent?
Non-empty mutually exclusive events are not independent. If one occurs, the other becomes impossible, so its conditional probability changes to zero. Independence instead means that learning one event occurred leaves the probability of the other unchanged.
What does the Central Limit Theorem say?
For a sufficiently large random sample, the sampling distribution of the mean becomes approximately normal even when the population is not normal. The required size depends on population shape. The theorem concerns repeated sample means, not the shape of raw observations.
Exam move
Draw a tree whenever the question contains given, sensitivity, specificity or a base rate. Convert percentages into counts for a convenient population, label every branch and total only the cases that satisfy the condition. Practise swapping the order of two conditionals and explain why the numerator remains the same while the denominator changes.
For independence, compute both the conditional form and the product rule so the concept is not reduced to a memorised phrase. For sampling distributions, say aloud what each point on the horizontal axis represents. If the points are sample means, do not describe them as individual observations. Use square-root scaling drills: double, quadruple and multiply sample size by nine, then state the effect on standard error.
Keep design and precision separate by pairing a large biased sample with a smaller probability sample. Finish each practice set by explaining why a tighter sampling distribution cannot repair coverage or measurement bias.
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