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Cumulative binomial probabilities and tables

Edexcel IAL Mathematics · WST02 Statistics 2

Most binomial questions in S2 ask for P(X ≤ x), P(X ≥ x) or a range. The cumulative tables, or a calculator, give P(X ≤ x) directly; everything else is built from it.

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What the tables give

The cumulative binomial table gives P(X ≤ x) to 4 decimal places for chosen values of n and p (with p up to 0.5). A calculator's binomial cumulative function gives the same values. Write down the probability statement before you look anything up.

Rewriting probabilities

For a discrete variable, '<' and '≤' differ:

P(X < x) = P(X ≤ x - 1)
P(X ≥ x) = 1 - P(X ≤ x - 1)
P(X > x) = 1 - P(X ≤ x)
P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a - 1)

You want	Write it as
P(X < 6)	P(X ≤ 5)
P(X ≥ 8)	1 - P(X ≤ 7)
P(X > 4)	1 - P(X ≤ 4)
P(3 ≤ X < 9)	P(X ≤ 8) - P(X ≤ 2)

When p is more than 0.5

Count failures instead. If X ~ B(15, 0.7) is the number of successes, then Y = 15 - X ~ B(15, 0.3) is the number of failures. P(X ≥ 12) = P(Y ≤ 3).

Using the tables backwards

Some questions give a probability and ask for a value. For X ~ B(20, 0.3), the smallest k with P(X ≤ k) > 0.95 is found by scanning the table: P(X ≤ 8) = 0.8867, P(X ≤ 9) = 0.9520, so k = 9.

Worked example

X ~ B(20, 0.3). Find P(3 ≤ X < 9).

P(3 ≤ X < 9) = P(X ≤ 8) - P(X ≤ 2)
= 0.8867 - 0.0355

Answer: 0.8512

Key idea: Tables give P(X ≤ x). Turn every other statement into that form, and count failures when p > 0.5.

Check yourself

X ~ B(20, 0.3). Find P(X ≤ 5). Give your answer to 4 decimal places.
Show answer
0.4164. Directly from the table: 0.4164.
X ~ B(20, 0.3). Find P(X ≥ 8). Give your answer to 4 decimal places.
Show answer
0.2277. 1 - P(X ≤ 7) = 1 - 0.7723 = 0.2277.
X ~ B(20, 0.3). Find P(3 ≤ X < 9). Give your answer to 4 decimal places.
Show answer
0.8512. P(X ≤ 8) - P(X ≤ 2) = 0.8867 - 0.0355 = 0.8512.
X ~ B(20, 0.3). Find P(X > 4). Give your answer to 4 decimal places.
Show answer
0.7625. 1 - P(X ≤ 4) = 1 - 0.2375 = 0.7625.
70% of a farm's eggs are large. In a box of 15 eggs, X is the number of large eggs. Find P(X ≥ 12). Give your answer to 4 decimal places.
Show answer
0.2969. Let Y = 15 - X ~ B(15, 0.3) count the eggs that are not large. P(X ≥ 12) = P(Y ≤ 3) = 0.2969.
X ~ B(20, 0.3). Find the smallest value of k such that P(X > k) < 0.05.
Show answer
9. P(X > k) < 0.05 means P(X ≤ k) > 0.95. P(X ≤ 8) = 0.8867 and P(X ≤ 9) = 0.9520, so k = 9.
For X ~ B(n, p), which is equal to P(X < 6)?
- P(X ≤ 6)
- P(X ≤ 5)
- 1 - P(X ≤ 6)
- 1 - P(X ≤ 5)
Show answer
P(X ≤ 5). X is a whole number, so X < 6 means X ≤ 5.

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Sources

Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2

Last updated 8 November 2026. Spotted a mistake? Tell us.