State the four conditions for a binomial model.

Fixed number of trials; two outcomes; constant p; independent trials.

What does \binom nr represent?

The number of ways of choosing r successes from n trials.

Why is (0.3)^3(0.7)^7 not P(X=3) for B(10,0.3)?

It is one order only; multiply by \binom{10}{3}.

Name a way in which independence might fail in a real situation.

People attending together, such as family members, or a machine drifting.

The binomial distributionAQA A-Level Maths: Flashcards

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Write the notation for a binomial distribution.

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Write the notation for a binomial distribution.: $X\sim B(n,p)$
State the four conditions for a binomial model.: Fixed number of trials; two outcomes; constant $p$ ; independent trials.
Formula for $P(X=r)$ ?: $\binom nrp^r(1-p)^{n-r}$
What does $\binom nr$ represent?: The number of ways of choosing $r$ successes from $n$ trials.
Rewrite $P(X<5)$ as a cumulative probability.: $P(X\le4)$
Rewrite $P(X\ge5)$ using $P(X\le r)$ .: $1-P(X\le4)$
Rewrite $P(X>5)$ using $P(X\le r)$ .: $1-P(X\le5)$
Rewrite $P(3\le X\le6)$ using $P(X\le r)$ .: $P(X\le6)-P(X\le2)$
What do all probabilities in a distribution add up to?: 1
How do you find $P(\text{at least one})$ in $n$ independent trials?: $1-(1-p)^n$
Why is $(0.3)^3(0.7)^7$ not $P(X=3)$ for $B(10,0.3)$ ?: It is one order only; multiply by $\binom{10}{3}$ .
Name a way in which independence might fail in a real situation.: People attending together, such as family members, or a machine drifting.
What should you do first when a binomial problem is in words?: Define $X$ , $n$ and $p$ .

Exam questions on The binomial distribution

A gardener plants 10 seeds. Each seed germinates with probability $0.3$ , independently of the others. The number of seeds that germinate is $X$ , where $X\sim B(10,0.3)$ .
State two conditions that must hold for $X$ to be modelled by a binomial distribution in this context.2 marks
A multiple-choice test has 20 questions, each with four options of which exactly one is correct. A student guesses every answer. The number of correct answers is $X$ , where $X\sim B(20,0.25)$ .
Find the probability that the student gets more than 7 questions correct.2 marks
A factory makes light bulbs and 8% of them are defective, independently of one another. A box contains 25 bulbs. The number of defective bulbs in a box is $Y$ , where $Y\sim B(25,0.08)$ .
Find the probability that a box contains exactly 2 defective bulbs.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).