Without replacement
Objective. Update conditional probabilities after an item is removed.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Without replacement describes situations where an item taken from a group is not put back, so the total number of items and the number of each type change for the next pick, altering conditional probabilities. On a tree diagram, the second set of branches must use updated totals that reflect the first selection's outcome. For example, picking a red ball from a bag of 5 red and 3 blue without replacing it leaves 4 red and 3 blue for the next pick, so the denominator decreases by 1. Always recalculate both the numerator and denominator for each subsequent branch.
Worked example
A bag has 4 red and 6 blue counters. Two are drawn without replacement. Find the probability both are red.
- 1P(first red) = 4/10
- 2After removing one red, remaining: 3 red, 9 total
- 3P(second red | first red) = 3/9
- 4P(both red) = (4/10) × (3/9) = 12/90 = 2/15
Answer: 2/15
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students often keep the first pick's probabilities unchanged for the second pick; without replacement always reduces the total count by one and adjusts whichever category was removed.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.