Q: What does IP Maths Notes (Lower Sec, Year 1-2): 10) Probability Models cover? A: Construct sample spaces, tree diagrams, and evaluate independent and dependent events with clear notation.
The core idea is simple: Probability is favourable outcomes divided by possible outcomes.
Use it as a working check: Use sample spaces for one-step events, tree diagrams for repeated events, and complements when "at least one" is easier to solve by finding "none".
Then go one layer deeper: Follow the counter, class, and dice examples to practise deciding whether replacement happens, multiplying along a path, and adding the allowed paths.
Probability measures uncertainty. This final post develops sample space construction, event notation, and multi-stage probability models.
These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).
Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.
Before calculating, decide which probability model matches the wording. This prevents you from using a simple fraction when the outcomes are not equally likely.
Question wording
Best first model
What to check
"One item is chosen from a list"
Sample space
Are all listed outcomes equally likely?
"Two or more stages happen"
Tree diagram
Does the probability change after each stage?
"At least one" or "none" appears
Complement
Is it shorter to find the opposite event first?
"Or" joins two events
Count or add allowed outcomes
Do the events overlap, so an outcome might be counted twice?
Common trap:totalfavourable only works directly when each outcome in the sample space is equally likely. If the question gives different probabilities for different branches, use those branch probabilities instead.
2 Sample spaces
Example: Toss two coins. Outcomes: HH,HT,TH,TT. Probability of exactly one head: 42=21.
3 Tree diagrams
Branch-denominator checkpoint
For multi-stage questions, update the total before writing the next branch probability.
Wording clue
What happens to the next denominator
Example branch update
"With replacement" or "put back"
Total stays the same each draw.
From 3 red and 2 blue: red then red is 53×53.
"Without replacement" or "not replaced"
Total decreases by 1 after each draw.
Red first changes the second red branch to 42.
"Given that" or "after knowing"
Work inside the reduced condition only.
If the first student is a girl, the second draw has 17 girls out of 29 students.
Common trap: do not copy the first fraction onto every second branch. Ask what has already been removed before writing the next denominator.
Tree path-combining checkpoint
After drawing a tree diagram, treat each complete route from left to right as one outcome path. Multiply along one path, then add the separate paths that satisfy the question.
Question asks for
Path work
Final combination
Both events happen
Use the single matching path.
Multiply the branch probabilities on that path.
Exactly one of two events happens
Use the first-event-only path and the second-event-only path.
Multiply along each path, then add the two answers.
At least one event happens
Either add all paths with one or more successes, or use the complement.
Usually faster: 1−P(no successes).
Event order matters
Keep the routes separate.
Red then blue and blue then red are different paths unless the question says order is ignored.
Worked check: if a red counter has probability 53 on each draw with replacement, then exactly one red in two draws is
53×52+52×53=2512.
Common trap: do not add branch probabilities before completing a path. A path such as red then blue needs multiplication first because both stages must happen.
Given-that filter checkpoint
When a question says "given that", the information after those words becomes the new allowed sample space. First filter to the outcomes that match the given information, then count the outcomes that also match the question.
Wording pattern
New denominator
Numerator to count
Trap to avoid
"Given that the counter is red or blue..."
Only red-or-blue outcomes
Outcomes in that group that satisfy the question
Keeping colours that the condition has already ruled out.
"Given that at least one child is a girl..."
All outcomes with one or more girls
Outcomes in that filtered group that match the target event
Using the original full sample space after the condition is known.
"Given that the first draw is red..."
Only paths starting with red
Paths among those red-first paths that satisfy the question
Multiplying by the probability of the given event again when it has already happened.
Worked check: a family has two children, and each child is equally likely to be a boy or a girl. The sample space is BB,BG,GB,GG. Given that at least one child is a girl, remove BB. The filtered sample space is BG,GB,GG, so
P(both girls∣at least one girl)=31.
Misconception check: 41 answers "both children are girls" before any extra information. Once the question says at least one child is a girl, the denominator is no longer 4.
3.1 Independent events
A bag contains 3 red and 2 blue counters. Replace after each draw.
Probability tree uses same probabilities at each level: P(R)=53, P(B)=52.
Probability of RR: 53×53=259
3.2 Without replacement (dependent)
If counters are not replaced, second-level probabilities change.
First draw red, second draw red: 53×42=103.
First draw blue, second draw red: 52×43=103
Worked example - Conditional probability
A class has 12 boys and 18 girls. Two students are selected without replacement. Find probability both are girls.
P(GG)=3018×2917=870306=14551.
4 Complementary reasoning
Sometimes it is easier to calculate the probability of an event not happening.
Example: Probability that at least one six appears in four fair dice rolls.
P(no six)=(65)4,P(at least one six)=1−(65)4.
Overlap checkpoint for "or" questions
When a question asks for A or B, check whether one outcome can satisfy both conditions. If there is overlap, subtract it once.
If the events cannot happen together, add the two probabilities: P(A or B)=P(A)+P(B).
If the events can overlap, subtract the overlap: P(A or B)=P(A)+P(B)−P(A and B).
In a counting question, count the overlap directly before writing the fraction.
Worked check: from tickets numbered 1 to 20, find the probability of drawing a multiple of 3 or 5. Multiples of 3 are 3,6,9,12,15,18, multiples of 5 are 5,10,15,20, and 15 is counted in both lists. Favourable outcomes =6+4−1=9, so the probability is 209.
Common trap: the word "or" in probability usually means either condition, including outcomes that satisfy both. Do not double-count the overlap.
5 Expected value preview
For a simple game with payouts xi and probabilities pi, expected value E(X)=∑pixi. Lower-sec exams rarely ask for it directly, but the concept frames decision-making in games of chance.
Practice Quiz
Consolidate tree-diagram reasoning, complements, conditional probabilities, and union logic with the quiz below.
Try it yourself
A box contains 4 green, 3 yellow, and 5 red pens. Two pens are drawn without replacement. Find the probability both are red.
The probability of a machine producing a defective item is 0.08. What is the probability that in a batch of three items, at least one is defective?
Two dice are rolled. Find the probability the sum is at least 10.
A bag contains tickets numbered 1 to 20. One ticket is drawn at random. What is the probability it is a multiple of 3 or 5?
