PSYB06H3 Chap.3 Probability Rules and Bayesian Reasoning
Probability Rules and Bayesian Reasoning
Three operations behind every question
Probability problems in this course are assembled from three moves. A joint probability asks for two outcomes together, an additive probability asks for one outcome or another, and a conditional probability asks for an outcome once another is known.
Recognising which move a question wants is usually the hardest step; the arithmetic that follows is short.
Overlap and exclusivity
When two outcomes can happen together, adding their probabilities double counts the shared part, so the overlap is subtracted once. Outcomes that cannot co-occur are mutually exclusive and have nothing to subtract.
A set of outcomes that covers every possibility is exhaustive and sums to one, which makes it possible to find a probability by subtracting its complement from one.
Independence and the multiplicative rule
Two outcomes are independent when knowing one leaves the probability of the other unchanged. Only then does the joint probability equal the simple product of the two probabilities.
In general the joint probability multiplies the first probability by the conditional probability of the second, a rule that works whether or not the outcomes are independent.
Comparing a table cell with the product of its margins is a quick independence test, and a mismatch tells you to keep the conditional form of the rule for any later calculation.
Updating with evidence
Bayes' rule reverses a conditional probability. It combines how common a hypothesis is beforehand, how often the evidence appears when the hypothesis is true, and how often the evidence appears overall.
Screening problems show why this matters: when a condition is rare, most positive results can come from people without it. Converting the percentages into counts for an imagined population of 1000 keeps the reasoning concrete and every total checkable.
Empirical and analytical answers
An analytical probability counts qualifying outcomes against all possible outcomes, such as three odd faces out of six on a die.
An empirical probability comes from repeating the event and counting what happens. With few repetitions the two can disagree noticeably because of random noise, but with many repetitions the empirical value settles close to the analytical one, much as larger samples better represent a population.
When a situation is too complex to analyse directly, repeated observation or simulation becomes the only practical route to a probability.
What this chapter covers
- 01
Analytical and Empirical Probability
- 02
Joint Probability
- 03
Additive Rule and Overlap
- 04
Mutually Exclusive and Exhaustive Outcomes
- 05
Conditional Probability and Independence
- 06
Bayes' Rule and Contingency Tables
Worked example · free
Sleep and anxiety in a contingency table
- 1Anxiety total: 48 + 60 = 108, so p(anxiety) = 108 / 300 = 0.36.
- 1Joint: p(insomnia and anxiety) = 48 / 300 = 0.16.
- 1Additive: 120/300 + 108/300 − 48/300 = 180/300 = 0.60.
- 1Conditional: 48 / 120 = 0.40, which differs from 0.36, so the reports are not independent.
Key terms
- Joint Probability
- The probability that two outcomes occur together, written p(A and B).
- Additive Rule
- The rule that p(A or B) equals p(A) plus p(B) minus their joint probability.
- Mutually Exclusive Events
- Outcomes that cannot occur together, so their joint probability is zero.
- Exhaustive Set
- A collection of outcomes that covers every possibility, so their probabilities sum to one.
- Independence
- The property that one outcome leaves the probability of another unchanged, so p(B given A) equals p(B).
- Posterior Probability
- The probability of a hypothesis after the evidence is taken into account, calculated with Bayes' rule.
Probability Rules and Bayesian Reasoning FAQ
How can I tell an AND question from a GIVEN question?
An AND question treats both outcomes as unknown and asks how often they occur together, so it divides by everyone. A GIVEN question tells you one outcome has already happened, so it divides only by that group. Rewording the question as a fraction of a specific group usually reveals which one is meant.
Are mutually exclusive events independent?
No. If two outcomes cannot happen together, learning that one occurred drops the probability of the other to zero, which is a change. Independence requires the probability to stay the same. The two ideas are often confused because both make the arithmetic simpler in different ways.
Why does a positive screening result often mean a low chance of the condition?
When a condition is rare, the many people without it generate false positives that can outnumber the true positives from the few people with it. The probability of the condition after a positive result depends on how common it was to begin with, not only on how accurate the test is.
Does a long run of one outcome make the other outcome due?
Not for independent events such as tosses of a fair coin. Each toss keeps the same probability regardless of earlier results, so eight heads in a row still leave the next toss at a probability of 0.50. Expecting a correction treats separate trials as if they remembered one another.
Exam move
Build a two-by-two contingency table for an everyday example and compute every joint, marginal and conditional probability it contains. Check independence by comparing a cell with the product of its margins. Then rework a Bayes problem twice, once with probabilities and once with counts for 1000 people, and confirm both routes agree to four decimal places.
Write down which group is the denominator before each division, and finish by stating in words what the final probability describes.
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