Binomial cumulative probabilities

The idea

Cumulative probabilities $P(X \le x)$ come from tables or the calculator's binomial CD. Turn every phrase into $\le$ form: $P(X \ge 5) = 1 - P(X \le 4)$ and $P(X > 5) = 1 - P(X \le 5)$ .

Translate the words: at least, more than, fewer than, between.
Rewrite using $P(X \le \dots)$ only.
Look up or calculate, then subtract.

Watch out 'At least $5$ ' is $1 - P(X \le 4)$ , not $1 - P(X \le 5)$ .

See it worked

Worked example

$X \sim B(20, 0.4)$ . Find $P(X \le 6)$ .

Use the binomial CD function with $n = 20$ , $p = 0.4$ , $x = 6$ .
$P(X \le 6) = 0.25001\ldots$
$= 0.2500$ (4 d.p.).

Answer $0.2500$

Your turn

Question 1Calculator allowed3 marks

$X \sim B(12, 0.5)$ . Find $P(X \ge 9)$ to 4 decimal places.

Question 2Calculator allowed3 marks

$X \sim B(15, 0.3)$ . Find $P(3 \le X \le 6)$ to 4 decimal places.

Question 3Calculator allowed3 marks

$X \sim B(10, 0.2)$ . Find $P(X > 2)$ to 4 decimal places.

In past papers

31 official questions in 27 papers since 2018. The latest:

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