For a binomial count X, ‘exactly 3’ means X = 3, while ‘at least 3’ includes 3 and all higher possible counts. To find P(X ≥ 3) using a lower-tail calculation, subtract P(X ≤ 2) from 1.
The 2027 A-Level H2 Mathematics 9758 syllabus covers choosing and using the binomial model. This example focuses on where the boundary goes when you turn a worded probability into a sum or a complement.
- Check that the trials are independent and share a constant success probability.
- Write the counts that the wording includes.
- For ‘at least k’, subtract the probability of k − 1 or fewer.
- Check whether the threshold itself belongs in the event.
CHAPTER 1 OF 4 · Back to contents
Set up the count before the event
Use a binomial model when you count successes across a fixed number of independent trials, each with the same probability of success. A trial has two outcomes for the purpose of the count. State what success means in context.
If there are six trials and X counts successes, X can take integer values from 0 through 6. Writing the possible values helps place the boundary. ‘At least 3’ retains 3, 4, 5 and 6. Its complement retains 0, 1 and 2.
A cumulative calculation P(X ≤ 3) includes the count 3. Subtracting that probability from 1 gives strictly more than 3.
CHAPTER 2 OF 4 · Back to contents
Worked example | Six checks with success probability 0.25
Six independent checks each give a positive result with probability 0.25. Let X be the number of positives. Then X has distribution B(6, 0.25). Compare exactly three, at least three and more than three positives.
| Wording | Included counts | Calculation |
|---|---|---|
| Exactly three | 3 | P(X = 3) |
| At least three | 3, 4, 5, 6 | 1 − P(X ≤ 2) |
| More than three | 4, 5, 6 | 1 − P(X ≤ 3) |
For exactly three, choose the three positive checks in 20 ways. The probability is 20(0.25)³(0.75)³ ≈ 0.1318. The other three checks are negative in every term of this calculation.
For at least three, add the probabilities of 0, 1 and 2 and subtract their sum from 1. This gives about 0.1694. For more than three, also subtract the probability of exactly three, giving about 0.03760.
The difference between the two upper-tail probabilities is exactly P(X = 3), before rounding. This is a quick check on which inequality you used.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise with a lower threshold
In the same model, find the probability of at least one positive. The complement is zero positives, so the answer is 1 − (0.75)⁶ ≈ 0.8220.
Now find the probability of at most one positive. Add P(X = 0) and P(X = 1), giving (0.75)⁶ + 6(0.25)(0.75)⁵ ≈ 0.5339. The complement of at most one is two or more.
In the exam, put an event such as P(X ≥ 1) on the page before using a calculator. If the working asks for a complement, write the last included count in the lower tail.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Is at least three the same as more than two?
Yes for an integer count. Both begin at three.
When would sampling without replacement change the model?
If the chance of success changes after a draw, the independent trials with constant success probability assumption generally fails. Check the stated sampling conditions.
Can I sum the upper tail directly?
Yes. Summing the individual probabilities from the threshold to the highest possible count gives the same answer.
Contents · Previous chapter · Continue with the related guide