Probability Diagrams and Formal Independent/Conditional Notation
Probability has its own diagrams and its own shorthand. Learn to read sample space diagrams, tree diagrams and Venn diagrams, and to use the formal notation for complement, independent, mutually exclusive and conditional events.
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The language of chance
Probability comes with its own diagrams and its own shorthand. Master both and long questions become quick. This module covers the diagrams - sample space diagrams, tree diagrams and Venn diagrams - and the formal notation for the complement, independent, mutually exclusive and conditional events. (Symbols shown here are standard exam notation.)
Probability words
Four ideas run through the whole topic.
Match each symbol to its meaning
- P(A')
- P(A ∩ B)
- P(A ∪ B)
- P(A | B)
- The probability that A does NOT happen
- The probability that both A and B happen
- The probability that A or B (or both) happens
- The probability of A given that B has happened
Independent events
For two independent events A and B, what is P(A and B)?
- P(A) multiplied by P(B)
- P(A) plus P(B)
- P(A) minus P(B)
- P(A) divided by P(B)
Three rules to know
Almost every probability calculation uses one of these: - Independent events: P(A ∩ B) = P(A) x P(B) (multiply) - Mutually exclusive events: P(A ∪ B) = P(A) + P(B) (add) - Conditional probability: P(A | B) = P(A ∩ B) / P(B) (divide) Choosing the right rule is the whole skill: multiply for independent "and", add for mutually exclusive "or", divide for "given".
Both heads
Two fair coins are flipped, which are independent events. What is the probability of getting heads on BOTH? Use P(A and B) = P(A) x P(B), with each probability 0.5.
Two ideas people mix up
Independent and mutually exclusive are NOT the same thing - and they use different rules.
Match each diagram to what it shows
- Sample space diagram
- Two-way table
- Tree diagram
- Venn diagram
- A grid of all possible outcomes of two events
- Frequencies split across two categories at once
- Branches showing outcomes and probabilities in stages
- Overlapping circles showing what sets share
Which show probability?
Select the TWO diagrams used to represent probability in this topic.
- Tree diagram
- Venn diagram
- Pie chart
- Histogram
It might rain
The probability that it rains tomorrow is P(A) = 0.3. Using P(A') = 1 - P(A), what is the probability that it does NOT rain?
The notation in words
In probability notation, P(A') means the _____ of A, found as 1 - P(A). P(A ∩ B) is read 'A _____ B'. If two events are _____, then P(A ∩ B) = P(A) x P(B). If two events are mutually _____, they cannot both happen. P(A | B) is a _____ probability.
Given that...
From a survey, P(A ∩ B) = 0.2 and P(B) = 0.5. Using P(A | B) = P(A ∩ B) / P(B), find P(A | B).
Pick the right rule
Three probability situations. Choose the correct approach for each.
- You flip a coin and roll a die. These are independent. How do you find P(heads AND a six)?
- On one roll of a die you want P(rolling a 2 OR a 5). These outcomes are mutually exclusive. What do you do?
- You want the probability someone owns a dog GIVEN that they own a cat. Which is the right tool?
Multiply for and, add for or
Winning marks here is about choosing the right rule: multiply for independent "and", add for mutually exclusive "or", and divide for a conditional "given". Keep two traps in mind: the complement is 1 - P(A), and mutually exclusive (cannot both happen) is NOT the same as independent (one does not affect the other). Read the words, pick the rule, then calculate.