What does \binom nr count?

The ways of choosing which r trials are successes

What do binomial tables give?

Cumulative probabilities P(X\le r)

How do you find p from mean and variance?

Divide: \frac{variance}{mean}=1-p

Name one situation where independence fails.

Sampling without replacement from a small population

The binomial distributionEdexcel International A Level Further Maths: Flashcards

Card 1 of 130 of 13 known

Question

Four conditions for a binomial model?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

Four conditions for a binomial model?: Fixed $n$ , two outcomes, constant $p$ , independent trials
Notation for a binomial distribution?: $X\sim\mathrm{B}(n,p)$
Formula for $\mathrm{P}(X=r)$ ?: $\binom nr p^r(1-p)^{n-r}$
What does $\binom nr$ count?: The ways of choosing which $r$ trials are successes
What do binomial tables give?: Cumulative probabilities $\mathrm{P}(X\le r)$
$\mathrm{P}(X\ge r)$ in terms of tables?: $1-\mathrm{P}(X\le r-1)$
$\mathrm{P}(a\le X\le b)$ in terms of tables?: $\mathrm{P}(X\le b)-\mathrm{P}(X\le a-1)$
What do you do for $p>0.5$ when tables stop at $0.5$ ?: Use $Y=n-X\sim\mathrm{B}(n,1-p)$
Mean of $\mathrm{B}(n,p)$ ?: $np$
Variance of $\mathrm{B}(n,p)$ ?: $np(1-p)$
How do you find $p$ from mean and variance?: Divide: $\frac{\text{variance}}{\text{mean}}=1-p$
How does the variance of a binomial compare with its mean?: It is smaller, since $1-p<1$ .
Name one situation where independence fails.: Sampling without replacement from a small population

Exam questions on The binomial distribution

The discrete random variable $X$ has the binomial distribution $\mathrm{B}(12,\,0.25)$ .
Find $\mathrm{P}(X\ge2)$ .2 marks
A machine makes components. Each component is independently defective with probability $0.04$ . A random sample of $20$ components is taken, and the number of defective components in the sample is $X$ .
Find the probability that more than $2$ components in the sample are defective.2 marks
A multiple-choice test has $15$ questions. Each question has four options, of which exactly one is correct. A student chooses an option at random for every question, independently. The number of correct answers is $X$ .
Find the probability that the student gets exactly $5$ questions correct.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).