A-Level Maths | Binomial Distribution
A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.
$X \sim B(n, p)$ means we are modelling the distribution $X$ binomially, where:
$n$: the number of trials
$p$: the probability of success on each trial
If $X \sim B(n,p)$, then $P(X=r) = \dbinom{n}{r}p^r(1-p)^{n-r}$
Worked Example
$X \sim B(10, 0.3)$. Find $P(X=4)$.
$P(X=4) = \dbinom{10}{4}(0.3)^4(0.7)^6 = 0.2001$
Question 1
$X \sim B(20, 0.4)$
a) Find $P(X=3)$
b) Find $P(X=6)$
c) Find $P(X=9)$
d) Find $P(X=12)$
e) Find $P(X=15)$
Show Answers
a) $0.0123$
b) $0.1244$
c) $0.1597$
d) $0.0355$
e) $0.0013$
Question 2
$X \sim B(25, 0.35)$
a) Find $P(X \leq 4)$
b) Find $P(X \leq 7)$
c) Find $P(X \leq 9)$
d) Find $P(X \leq 12)$
e) Find $P(X \leq 15)$
Show Answers
a) $0.0320$
b) $0.3061$
c) $0.6303$
d) $0.9396$
e) $0.9971$
Worked Example
$X \sim B(18, 0.25)$. Find $P(X < 7)$.
$P(X < 7) = P(X \leq 6) = 0.8610$
Question 3
$X \sim B(30, 0.3)$
a) Find $P(X < 5)$
b) Find $P(X < 8)$
c) Find $P(X < 10)$
d) Find $P(X < 13)$
e) Find $P(X < 16)$
Show Answers
a) $0.0302$
b) $0.2814$
c) $0.5888$
d) $0.9155$
e) $0.9936$
Worked Example
$X \sim B(18, 0.25)$. Find $P(X \geq 7)$.
$P(X \geq 7) = 1 - P(X \leq 6) = 1 - 0.8610 = 0.1390$
Question 4
$X \sim B(20, 0.4)$
a) Find $P(X \geq 5)$
b) Find $P(X \geq 8)$
c) Find $P(X \geq 10)$
d) Find $P(X \geq 13)$
e) Find $P(X \geq 16)$
Show Answers
a) $0.9490$
b) $0.5841$
c) $0.2447$
d) $0.0210$
e) $0.0003$
Left tail hypothesis testing
Question 5
Question 6
Question 7
Question 8
Right tail hypothesis testing
Question 9
Question 10
Question 11
Question 12
Two-tail hypothesis testing
Question 13
Question 14
Left tail hypothesis test critical values
Left tail
Question 15
Right tail hypothesis test critical values
Question 16
Two-tail hypothesis test critical values
Question 17
Question 18
Question 19
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