S6.3StatisticsStretch

Cumulative Probabilities

Calculating P(X = r) for many values is slow. Cumulative binomial probabilities, $P (X \leq r)$ , found from a calculator or tables, let you answer “at most”, “fewer than”, “at least” and “more than” questions quickly.

25 min Video by Zeeshan Zamurred Statistical Distributions

Edexcel AS Level Maths: 6.3 Cumulative ProbabilitiesWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand cumulative probability P(X ≤ r)
Convert worded phrases into inequalities
Use the complement for “at least”/“more than”
Find ranges with subtraction

Cumulative probability

A cumulative probability $P (X \leq r)$ is the chance of $r$ $or fewer$ successes. Calculators and tables give these directly, so you rarely need to add many individual terms.

P (X \leq r) = P (X = 0) + P (X = 1) + \dots + P (X = r)

The running total up to

r

Translating the words

Exam wording maps onto inequalities: “at most $r$ ” is $P (X \leq r)$ ; “fewer than $r$ ” is $P (X \leq r - 1)$ ; “at least $r$ ” is $1 - P (X \leq r - 1)$ ; “more than $r$ ” is $1 - P (X \leq r)$ .

P (X \geq r) = 1 - P (X \leq r - 1)

Use the complement for “at least / more than”.

Tip — Watch the boundary: “fewer than r” excludes r, so it is P(X ≤ r − 1).

Ranges by subtraction

For a range like $P (a \leq X \leq b)$ , subtract cumulative values: $P (X \leq b) - P (X \leq a - 1)$ .

P (X \leq 7) - P (X \leq 2)

Answer

P (X \leq 7) - P (X \leq 2)

Formula recap

P (X \leq r)

Cumulative: r or fewer.

P (X \geq r) = 1 - P (X \leq r - 1)

“At least” via complement.

P (a \leq X \leq b) = P (X \leq b) - P (X \leq a - 1)

Range by subtraction.

Common mistakes to avoid

Reading “at least r” as P(X ≤ r).

“At least r” is P(X ≥ r) = 1 − P(X ≤ r − 1).

Off-by-one errors with “fewer than” / “more than”.

“Fewer than r” = P(X ≤ r − 1); “more than r” = 1 − P(X ≤ r).

Key takeaways

P(X ≤ r) is the cumulative probability of r or fewer.
“At least r” = 1 − P(X ≤ r − 1); “more than r” = 1 − P(X ≤ r).
Ranges: P(X ≤ b) − P(X ≤ a − 1).

Test yourself

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