Probability problems become manageable when you translate the wording into events, identify what is known and choose the matching rule. Many errors happen because students confuse independent events with mutually exclusive events, or use the unconditional probability when the question gives new information. This guide walks through the core ideas with small examples.
Define the events before calculating
Name the events clearly, such as A = a randomly selected student passed the exam and B = the student attended a review session. Write down the probability information from the question and note whether the events are independent, mutually exclusive or conditional.
For a compound problem, draw a tree diagram, two-way table or Venn diagram if it helps organise the cases. A diagram can make the denominator and relevant branch easier to identify.
Use complements and the addition rule
The complement of event A is the event that A does not occur, so P(not A) = 1 - P(A). This is especially useful for questions that ask for 'at least one' event, because it may be easier to calculate the probability that none occur and subtract it from one.
For events A and B, the general addition rule is P(A or B) = P(A) + P(B) - P(A and B). If the events are mutually exclusive, the overlap is zero. Do not use the simplified mutually-exclusive version unless the problem supports it.
If P(A) = 0.30, then
P(not A) = 1 - 0.30 = 0.70
General addition rule:
P(A or B) = P(A) + P(B) - P(A and B)Conditional probability
Conditional probability asks for the probability of A given that B has occurred. Its formula is P(A given B) = P(A and B) / P(B), provided P(B) is greater than zero. The condition changes the relevant group or denominator, which is where many errors happen.
For example, if a table includes all students and the question says 'among students who attended the review session', use the attendees as the reference group rather than all students. Read phrases such as 'given that', 'among those who', and 'if we know' carefully.
Independent events and Bayes' theorem
Two events are independent when knowing that one happened does not change the probability of the other, under the model. For independent events, P(A and B) = P(A) × P(B). Mutually exclusive events cannot both happen; that is a different property and usually does not imply independence.
Bayes' theorem reverses a conditional probability using prior probability and the likelihood of observed evidence. It is useful when a problem gives a test's sensitivity and false-positive rate, or when you need P(A given B) but are given P(B given A). Always keep track of which event is in the condition.
How to check your final answer
A probability must fall between zero and one. Check that the denominator represents the correct sample or conditional group, and that a percentage has been converted to a decimal when needed. If events are arranged in a tree, the probabilities of all branches leaving the same node should sum to one.
Write one sentence interpreting the result. A correct decimal with no explanation may leave the reader unsure which event you calculated.
- Define each event in words.
- Highlight what is being conditioned on.
- Choose the addition, multiplication, conditional or Bayes rule that matches the event structure.
- Check whether independence is actually stated or justified.
- Confirm the result is between 0 and 1 and explain it in context.
Understanding the concept is the first step.
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