For understanding probability exercises
Binomial probability at least vs more than
At least k includes k. More than k starts at k + 1. Keep n and p the same, check which counts belong to your event, then calculate its probability below.
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The difference is exactly one success count
For a binomial random variable X, “at least k” means X≥k, and “more than k” means X>k. Because X is an integer count, more than k is the same as at least k + 1.
That one-count difference is the probability of exactly k. For the same n and p, P(X≥k) = P(X>k) + P(X=k).
| Wording | Symbol | Included counts |
|---|---|---|
| At least k | X ≥ k | k through n |
| More than k | X > k | k + 1 through n |
| At most k | X ≤ k | 0 through k |
| Less than k | X < k | 0 through k − 1 |
| Exactly k | X = k | k only |
Compare both events with the same numbers
This separate mathematical example uses n = 3, p = 0.5 and k = 2. The PMF weights for zero through three successes are 1/8, 3/8, 3/8, 1/8.
At least two successes
Include two and three successes. P(X≥2) = 3/8 + 1/8 = 0.5 = 50%.
More than two successes
Include three successes only. P(X>2) = 1/8 = 0.125 = 12.5%.
The difference is 0.5 − 0.125 = 0.375, which is P(X=2). Changing only the wording changes the probability; the trial count and per-trial chance have stayed the same.
Use the correct complement
Let F(k) = P(X≤k) be the binomial CDF. The complement must contain every excluded count and none of the included ones.
- For at least k, exclude 0 through k−1.
P(X≥k) = 1−F(k−1). - For more than k, exclude 0 through k.
P(X>k) = 1−F(k).
In the example, 1−F(1) = 1−0.5 = 0.5 gives at least two. Using 1−F(2) = 1−0.875 = 0.125 instead gives more than two.
Use the CDF calculator and cumulative table to inspect those left-tail values. For a tiny right tail, choose the right-tail event directly rather than subtracting a rounded displayed CDF from 1.
Less than and at most have the same boundary issue
Less than k excludes k, so P(X<k) = F(k−1). At most k includes k, so P(X≤k) = F(k).
With n = 3, p = 0.5 and k = 2, fewer than two has probability 0.5, while at most two has probability 0.875. The difference is again P(X=2).
“No more than” means at most. “No fewer than” means at least. Write the inequality down before entering the values. If you need help translating the rest of the question, follow the step-by-step guide.
Check zero and n explicitly
At least zero is certain. More than n is impossible. At least n means exactly n, which is different from more than n. These checks also apply at p = 0 or p = 1.
The calculator accepts k from 0 through n. Select Less than k with k = 0 to obtain the empty event, rather than entering a negative count. When reasoning with CDF notation outside the form, use F(−1) = 0.
The PMF and CDF definitions are given in NIST’s binomial reference. The identities here follow by partitioning the integer counts. This guide does not infer whether a written exercise actually meets the independence and constant-probability assumptions.
Check your own probability
Enter the values from your question. Pick its wording, confirm the assumptions and calculate. You can then open the working, statistics, table or graph. Editing a value clears the previous result.
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Find the explanation you need
Exactly k successes, with combinations and formula steps.
Cumulative probability, tail boundaries and inverse cutoffs.
Read the question, choose the event and check each step.
All tools use 0 ≤ n ≤ 1000 and decimal p from 0 to 1. They do not choose a model or interpret a written problem automatically. Read privacy, assumptions and use limits.