To find the probability of at least one red counter in two draws, calculate the probability of no red counters and subtract it from one. If the first counter is kept out, update the number of counters in the bag before finding the second probability.
The 2027 SEC G3 Mathematics K310 syllabus covers combined events and tree diagrams. This guide compares two versions of the same problem so you can see exactly which factor changes when a counter is replaced.
- Translate “at least one” into its opposite.
- Follow the branch with no successes.
- Update the bag for the second draw.
- Subtract the probability of that branch from one.
CHAPTER 1 OF 4 · Back to contents
Find the event that has only one branch
A bag contains four red counters and six blue counters. Two counters are drawn at random, without replacement. “At least one red” includes red then blue, blue then red, and red then red. The opposite event is blue then blue.
CHAPTER 2 OF 4 · Back to contents
Worked example | Keep the first counter out
The probability of drawing blue first is 6/10. If the first counter is blue and stays out, five blue counters remain among nine counters. The probability of two blues is therefore (6/10) × (5/9) = 1/3. The probability of at least one red is 1 − 1/3 = 2/3.
To check with the other branches, calculate (4/10) × (6/9) for red then blue, (6/10) × (4/9) for blue then red, and (4/10) × (3/9) for two reds. The sum is 2/3.
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CHAPTER 3 OF 4 · Back to contents
Change the instruction to “with replacement”
Return the first counter and mix the bag before the second draw. The probability of blue on the second draw is then 6/10 regardless of the first result. The probability of at least one red becomes 1 − (6/10)² = 16/25.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Does at least one mean exactly one?
At least one includes two reds in a two-draw question. Exactly one includes only red–blue and blue–red.
What can a parent ask during revision?
Ask the student how many blue counters and how many counters in total are left after drawing a blue. Then ask which fraction uses both of those counts.