A reliable method
For successive selections, update the counts when an item is not replaced. Multiply along a particular path and add mutually exclusive alternatives when needed.
The common trap
After selecting a red counter without replacement, both the red count and total count fall by one. Independent events do not change each other’s probability.
Make the connection
Expected frequency is a long-run average prediction, not a guarantee for one experiment. Check every probability lies between zero and one.
A second draw may change the denominator
With three red and two blue counters, the chance of red on the first draw is 3/5. Without replacement, a red first draw leaves two red out of four, so the chance of two reds is (3/5)(2/4)=3/10. With replacement, the second red chance stays 3/5 and the joint probability is 9/25. The difference is the experimental procedure, not a different multiplication rule. For an event such as exactly one red, add the mutually exclusive red-blue and blue-red paths after multiplying along each path.
Transfer, explain and check
For the same bag without replacement, exactly one red has probability (3/5)(2/4)+(2/5)(3/4)=3/5. Explain why using only the first path gives half the required probability.
A bag contains 3 red and 2 blue counters. Two counters are drawn uniformly at random without replacement. Find the probability both are red.
- The first red probability is 3/5.
- After red is removed, the second is 2/4.
- Multiply the two probabilities along this path.
Compare the other answer choices
Reduce total count but forget a red was removed.
Use replacement probabilities for both draws.
Use only the first-draw red probability.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
A bag has 3 red and 2 blue counters; draw two without replacement. (a) find P(two reds); (b) find P(exactly one red), including both orders; (c) compare P(two reds) with replacement and explain the difference.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Cambridge IGCSE Mathematics 0580