Secondary 3 G2 Mathematics Tutor Bukit Timah | Probability Without Replacement: Tree Diagrams. Did you know? The first draw changes the second draw when a counter is not returned.
This Secondary 3 G2 Mathematics tutor guide for Bukit Timah families explains probability without replacement: tree diagrams with worked examples, checks and realistic limits. Read the quick answer first, then use the relevant section to repair a specific misunderstanding.
Quick answer
Suppose a bag has three red counters and two blue counters. Two counters are drawn sequentially without replacement. The total begins at five and becomes four after the first draw. The chance of red first is 3/5, and blue first is 2/5. These branches sum to one.
Find the exact question
Read the experiment · First-branch probabilities · Second draw after red · Second draw after blue · Two reds · Two blues · Exactly one red · At least one red · Compare replacement · Tree audit
Read the experiment
Suppose a bag has three red counters and two blue counters. Two counters are drawn sequentially without replacement. The total begins at five and becomes four after the first draw.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
First-branch probabilities
The chance of red first is 3/5, and blue first is 2/5. These branches sum to one.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Second draw after red
If red was first, two red and two blue remain. The next probabilities are red 2/4 and blue 2/4.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Second draw after blue
If blue was first, three red and one blue remain. The next probabilities are red 3/4 and blue 1/4.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Two reds
Multiply along the red-red path: (3/5)(2/4)=6/20=3/10.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Two blues
Multiply along blue-blue: (2/5)(1/4)=2/20=1/10.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Exactly one red
Red-blue has probability (3/5)(2/4)=3/10. Blue-red has (2/5)(3/4)=3/10. Add the distinct paths for 3/5.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
At least one red
Use the complement: one minus probability of two blues = 1−1/10=9/10.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Compare replacement
If counters are returned and mixed, red-red becomes (3/5)²=9/25. The second draw is no longer affected by the first.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Tree audit
All complete path probabilities should sum to one: RR=3/10, RB=3/10, BR=3/10 and BB=1/10. The sum is one.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A seven-day revision routine
Day 1: identify the key definition and the given quantities. Day 2: reproduce a worked example with reasons. Day 3: change the numbers. Day 4: solve an unlabelled problem. Day 5: diagnose the first error. Day 6: combine the idea with another topic. Day 7: retest independently after a delay.
Frequently asked questions
How should I start? Name the unknown and what the question actually asks. How can I verify the answer? Substitute it into the original information and check that all restrictions are met. Why can a correct-looking number be wrong? The model may have used the wrong reference, denominator or condition.
Official syllabus and tutorial context
Consult the 2027 SEC G2 Mathematics K210 syllabus for the exact assessment scope. The unchanged eduKateSG three-pupil Mathematics reference describes tutorials near Sixth Avenue. Confirm the current tutor, class fit, fees, venue and timetable directly.
Further Mathematics revision
Related resources: Quadratic Graphs, Standard Form and Accuracy and Probability Rules.