Calculate probabilities from binomial distribution

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

{pre} $\var{thismany}$ {post}

{something} $\var{number1}$ {else}

Give any introductory information the student needs.

			Name	Type	Generated Value

pre	string
descx1	string	number of chocolate chip cooki
something	string
thisnumber	integer	4
what	string	daily sales.
things	string	chocolate chip cookies.
descx	string	the number of chocolate chip c
tol	number	0.001
prob	rational	4/25
thisaswell	string	our selection contains no more
else	string	biscuits are selected at rando
thismany	integer	16
number1	integer	10
post	string	% of biscuits made by a baker
prob2	decimal	dec("0.794")
prob1	number	0.048
thatnumber	integer	2
this	string	our selection contains exactly
v	integer	1
tprob1	number	0.0483476024
tprob2	rational	5693796837063267/7174648137343063
sd	decimal	dec("1.159")

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Generated value: `string`

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Statement

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

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Assuming a binomial distribution for $X$ , {descX}, write down the values of $n$ and $p$ .

$X \sim \operatorname{bin}(n,p)$

$n=\;$ ?Gap 0. $p=\;$ ?Gap 1.

Find $\operatorname{E}[X]$ the expected {descX1}

$\operatorname{E}[X]=$ ?Gap 2.

Find the standard deviation for the {descX1}

Standard deviation = ? Gap 3. (to 3 decimal places).

Advice

1. $X \sim \operatorname{bin}(\var{number1},\var{prob})$ , so $n= \var{number1},\;\;p=\var{prob}$ .

2. The expectation is given by $\operatorname{E}[X]=n\times p=\var{number1}\times \var{prob}=\var{number1*prob}$

3. $\operatorname{stdev}(X)=\sqrt{n\times p \times (1-p)}=\sqrt{\var{number1}\times \var{prob} \times \var{1-prob}}=\var{sd}$ to 3 decimal places.

$\begin{eqnarray*}\operatorname{P}(X = \var{thisnumber}) &=& \dbinom{\var{number1}}{\var{thisnumber}}\times\var{prob}^{\var{thisnumber}}\times(1-\var{prob})^{\var{number1-thisnumber}}\\& =& \var{comb(number1,thisnumber)} \times\var{prob}^{\var{thisnumber}}\times\var{1-prob}^{\var{number1-thisnumber}}\\&=&\var{prob1}\end{eqnarray*}$ to 3 decimal places.

$\begin{eqnarray*}\operatorname{P}(X \leq \var{thatnumber})& =& \simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\& =& \simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})}\\& =& \var{prob2}\end{eqnarray*}$

to 3 decimal places.

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Calculate probabilities from binomial distribution

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Taxonomy: mathcentre

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `string`

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Gap-fill

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Error

Calculate probabilities from binomial distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`