Conditional probability
Section: Probability | Syllabus: Cambridge IGCSE Mathematics (0580)
Conditional Probability from Two-Way Tables
Extended Only
- Conditional probability is the probability of an event, given that another event has already happened; the word "given" restricts attention to only part of the data
- The notation P(A\,|\,B) means "the probability of A, given that B has happened"
- From a two-way table: restrict attention to only the row or column matching the given condition, then use that row/column's own total as the denominator - not the grand total of the whole table
Worked Example: Reading Conditional Probability from a Two-Way Table
- Question: A survey of 60 students records whether they play a musical instrument and whether they play a sport.
A student is chosen at random. Find the probability that the student plays a sport, given that they play a musical instrument.Plays a sport Does not play a sport Total Plays an instrument 12 8 20 No instrument 25 15 40 Total 37 23 60 - Step 1: "Given that they play a musical instrument" restricts attention to the "Plays an instrument" row only, which has a row total of 20
- Step 2: Of these 20 students, 12 also play a sport
- Answer: P(sport\,|\,instrument)=(12/20)=(3/5)
Common Mistakes
MistakeUsing the grand total of the whole table (60) as the denominator instead of the restricted row or column total
Fixthe denominator for a conditional probability is always the total of just the row or column named by the "given" condition
MistakeMuddling up which event is "given" and which is being found, and restricting the wrong row or column
Fixthe event written after the vertical bar in P(A\,|\,B) is always the one that has already happened - restrict to that row/column first
Conditional Probability from Tree Diagrams
Extended Only
- On a tree diagram where items are picked without replacement, the second-stage branch probabilities are already conditional probabilities - they assume the first pick has already happened, so the remaining totals have changed
- General rule linking combined and conditional probability: P(A and B)=P(A)× P(B\,|\,A)
- Rearranging: P(B\,|\,A)=(P(A and B)/P(A))
Worked Example: Conditional Probability on a Tree Diagram
- Question: A box contains 4 blue and 3 black pens. Two pens are taken at random without replacement. Given that the first pen taken is black, find the probability that the second pen is also black.
- Step 1: "Given that the first pen is black" means one black pen has already been removed, leaving 6 pens in total
- Step 2: Of these 6 remaining pens, only 3-1=2 are black
- Answer: P(second black\,|\,first black)=(2/6)=(1/3)
Common Mistakes
MistakeUsing the original totals (before the first pen was removed) instead of updating both the numerator and denominator
Fixonce the given condition has happened, both the count of that type and the overall total must be reduced by 1 before calculating the conditional probability
MistakeTrying to use the combined-probability formula P(A)× P(B) instead of reading the already-conditional branch value directly
Fixwhen a condition is given, the answer usually comes straight from one branch's probability, not from multiplying two branches together
Conditional Probability from Venn Diagrams
Extended Only
- On a Venn diagram, the number written in each region shows how many items belong there; the overlapping region shows items belonging to both sets
- Conditional probability from a Venn diagram: P(A\,|\,B)=(n(A and B)/n(B)) - the numerator is the count in the overlap, the denominator is the total count in the "given" set
Worked Example: Conditional Probability on a Venn Diagram
- Question: In a class of 30 students, a Venn diagram shows that 9 like only football, 6 like only basketball, 8 like both, and 7 like neither. A student is chosen at random. Find the probability that the student likes basketball, given that they like football.
- Step 1: The total who like football =9+8=17 (only football, plus both)
- Step 2: Of these, the number who also like basketball is the overlap, 8
- Answer: P(basketball\,|\,football)=(8/17)
Common Mistakes
MistakeUsing only the "football-only" region (9) as the denominator instead of the whole football total (both regions combined)
Fixthe "given" set includes the overlap as well as its own region on its own - add both parts of the circle together for the denominator
MistakeUsing the grand total of 30 students as the denominator
Fixthe grand total is only used for an unconditional probability - once a condition is given, the denominator shrinks to just that set's total
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