Further Mathematics · Pearson Edexcel 9FM0 · Option FS1 · Topic 3

Geometric and negative binomial distributions

The geometric and negative binomial distributions: how many trials until the first (or rth) success.

3 sub-topics
Taught in Year 13
49 practice questions
3 animated lessons

Practice questions (49) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

3.1Geometric and negative binomial modelsY13
3.2Mean and variance of the geometric distributionY13
3.3Mean and variance of the negative binomialY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 49 questions in this topic, free, no account needed.

Geometric and negative binomial distributions

Not needed for Year 12Easy

The geometric distribution · Year 13 · 1 mark

$X$ is the number of trials up to and including the first success, and each trial succeeds with probability $\tfrac{1}{6}$ . What is $\mathrm{E}(X)$ ?
(1)

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For a geometric distribution, $\mathrm{E}(X) = \dfrac{1}{p}$ . $D$ : $6$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Geometric and negative binomial distributions

Not needed for Year 12Medium

Geometric and negative binomial models · Year 13 · 7 marks

A basketball player takes shots from the free-throw line. Each shot succeeds with probability 0.4, independently.

(a)
Find the probability that her first success is on her 5th shot.
(2)
(b)
Find the probability that she needs more than 5 shots to get her first success.
(2)
(c)
Find the probability that her second success is on her 8th shot.
(3)

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Part (a)2 marks

Geometric: $k - 1$ failures, then a success. $(0.6)^{4}(0.4)$
$= 0.0518$

Part (b)2 marks

The first $k$ shots all fail. $\mathrm{P}(X > 5) = (0.6)^{5}$
$= 0.0778$

Part (c)3 marks

Negative binomial: 1 success in the first 7 shots, then a success. $\binom{7}{1}(0.4)^{2}(0.6)^{6}$
$= 0.0523$

Total7

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Geometric and negative binomial distributions

Not needed for Year 12Hard

Mean and variance of the negative binomial · Year 13 · 6 marks

The random variable

X

has a negative binomial distribution with mean 10 and variance 10.

(a)
Find the values of the parameters $r$ and $p$ .
(4)
(b)
Find $\mathrm{P}(X = 7)$ .
(2)

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Part (a)4 marks

$\frac{r}{p}$ ${}= 10, \ \frac{r(1 - p)}{p^2}$ ${}= 10$
Divide the variance by the mean. $\frac{1 - p}{p} = 1$
$p = 0.5$
$r = 5$

Part (b)2 marks

$\binom{6}{4}(0.5)^{5}(0.5)^2$
$= 0.1172$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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