What does X\sim B(n,p) mean?

X counts successes in n independent trials, each with probability p.

What does \binom nr count?

The ways of choosing which r of the n trials are successes.

What must you do when you divide an inequality by \log0.4?

Reverse the inequality sign, since \log0.4<0.

Give an example where independence fails for a binomial model.

Students from one class, who may travel or behave in the same way.

A probability found from one binomial can serve as p for another. When?

Two-stage problems, e.g. the number of accepted samples.

The binomial distributionEdexcel A-Level Maths: Flashcards

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Four conditions for a binomial model?

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Four conditions for a binomial model?: Fixed $n$ ; two outcomes; constant $p$ ; independent trials.
What does $X\sim B(n,p)$ mean?: $X$ counts successes in $n$ independent trials, each with probability $p$ .
Formula for $P(X=r)$ ?: $\binom nr p^r(1-p)^{n-r}$
What does $\binom nr$ count?: The ways of choosing which $r$ of the $n$ trials are successes.
What does the calculator's cumulative binomial function give?: $P(X\le r)$
Write $P(X\ge r)$ using a cumulative probability.: $1-P(X\le r-1)$
Write $P(X<r)$ using a cumulative probability.: $P(X\le r-1)$
Write $P(a\le X\le b)$ using cumulative probabilities.: $P(X\le b)-P(X\le a-1)$
Probability of at least one success in $n$ trials?: $1-(1-p)^n$
$X\sim B(5,0.5)$ : $P(X=0)$ ?: $0.5^5=0.03125$
What must you do when you divide an inequality by $\log0.4$ ?: Reverse the inequality sign, since $\log0.4<0$ .
Give an example where independence fails for a binomial model.: Students from one class, who may travel or behave in the same way.
A probability found from one binomial can serve as $p$ for another. When?: Two-stage problems, e.g. the number of accepted samples.

Exam questions on The binomial distribution

A student guesses the answer to every question on a 10-question multiple-choice test. Each question has four options, exactly one of which is correct. The random variable $X$ is the number of questions answered correctly, and $X\sim B(10,0.25)$ .
Find $P(X\ge4)$ .2 marks
A gardener plants 12 seeds. Each seed germinates with probability $0.9$ , independently of the others. The random variable $Y$ is the number of seeds that germinate.
Find $P(Y=10)$ .2 marks
A machine produces bolts. Each bolt is defective with probability $0.05$ , independently of the others. Bolts are packed in samples of 20, and $X$ is the number of defective bolts in a sample.
State the distribution of $X$ and find the probability that a sample contains exactly 2 defective bolts.3 marks

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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).