If the first trial is known to be a failure, it contributes zero successes. In a sequence of independent trials, the remaining trials still have their original success probability. Use a binomial model with one fewer trial.
Write down what the known trial contributes before using the calculator. If the first outcome is instead a success, it contributes one towards the total target.
- Record how many successes the known trial contributes.
- Model the remaining independent trials.
- Adjust the success target before calculating.
Worked example | First trial fails
There are four independent trials and the probability of success on each is ¼. The first trial is known to have failed. What is the probability of at least two successes in all four trials?
The remaining count Y follows B(3, ¼). The original target still requires at least two successes among those three trials. So P(Y ≥ 2) = 3 × (¼)² × ¾ + (¼)³ = 10/64 = 5/32.
Compare a known first-trial success
Keep the four-trial setup but suppose the first trial succeeds. The other three trials still follow B(3, ¼). This time you need only one more success to reach two in total.
The required probability is 1 − (¾)³ = 37/64. Both conditions leave three trials. What changes is how many more successes you need.
Try five independent trials
Each trial has a success probability of ½. Given the first trial fails, what is the probability of at least two successes in all five trials?
Answer. The remaining count is B(4, ½). Subtract the probabilities of zero and one success from one. The result is 1 − 1/16 − 4/16 = 11/16.
Frequently asked questions
Why does the success probability stay the same?
Independence means the known first outcome leaves the probabilities of the other trials unchanged.
Can I do the same with draws without replacement?
A draw without replacement generally changes the make-up of what remains. Work out the probabilities for the revised situation before choosing a model.
Official information
SEAB, 2027 GCE A-Level H2 Mathematics syllabus (9758), Probability and Binomial Distribution. Syllabus checked 11 November 2026. The examples are original.