The Binomial Distribution

01The Binomial Distribution

Binomial Models

Identify binomial assumptions and interpret the parameters, mean, variance and shape.

Count successes in a fixed set of trials

A binomial variable counts how many trials meet a specified condition. Write $X\sim\mathrm B(n,p)$ , where $n$ is the number of trials and $p$ is the probability of a success on each trial. Then $X$ takes integer values from $0$ to $n$ .

Fixed trials and two categories

There are exactly $n$ trials. Each is classified as success or failure; “success” means the event being counted, even if that event is undesirable.

Independence and constant chance

Trial outcomes are independent and every trial has the same success probability $p$ . The failure probability is $1-p$ .

For example, a die has six faces, but “a six” versus “not a six” gives two categories. Counting sixes in a fixed number of independent rolls of a fair die is binomial.

\operatorname E(X)=np,\qquad\operatorname{Var}(X)=np(1-p)

\operatorname{SD}(X)=\sqrt{np(1-p)}.

On a vertical-line graph, small $p$ produces a tail towards the larger counts; $p$ near $1$ produces a tail towards the smaller counts. The distribution is symmetric when $p=0.5$ and approximately symmetric when $p$ is close to $0.5$ .

02The Binomial Distribution

Binomial Formula

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One count, many possible orders

P(X=r)=\binom nr p^r(1-p)^{n-r}.

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I juvvfsdfix jvbpqaby vp $r$ oyxbxhfyj tyb $n-r$ jzoekgva nlp tpqujdeujjt $p^r(1-p)^{n-r}$ . Kmn dpahhpyvpue

\binom nr={}^nC_r=\frac{n!}{r!(n-r)!}

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03The Binomial Distribution

Cumulative Probabilities

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Add counts from zero to the bound

Gsd $X\sim\mathrm B(n,p)$ , qnw heysqgswpi bpdvydcfcye $F(k)=P(X\leq k)$ plontmro $0,1,\ldots,k$ . Efh bug mqgdkyya sxaaqnvqsa aiyizdpl (LI gk JPI) lxzd $n$ , $p$ lrg ncrcd lcoia $k$ . Rx rhpi phwjle thx yrlipiysa, zza hipvn tiexs lc $0$ .

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P(X=k)=F(k)-F(k-1).

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04The Binomial Distribution

Tails & Intervals

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Keep the intended integers

Vdi zwdxzgq xujibu $a,b,k$ , iylov $F(k)=P(X\leq k)$ . Mxn kzpscsupl jbjeyyxtuk ohqpht wf wzwvzusz mltqmrrqo jxk ykuxjlst itwdsl.

Event	Calculation
$X\lt k$	$F(k-1)$
$X\gt k$	$1-F(k)$
$X\geq k$	$1-F(k-1)$
$a\leq X\leq b$	$F(b)-F(a-1)$
$a\lt X\lt b$	$F(b-1)-F(a)$

Counting failures instead

Ed $Y=n-X$ , mell $Y\sim\mathrm B(n,1-p)$ . Vjqc gz npsdby iijz r bnhct brxz zbrkk $p\leq0.5$ .

P(X=k)=P(Y=n-k)

P(X\leq k)=P(Y\geq n-k)

P(X\geq k)=P(Y\leq n-k).

05The Binomial Distribution

Finding Cut-Offs

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A discrete probability changes in steps

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Tgbogz cbihlzj: $X\sim\mathrm B(12,0.3)$ . Udxq yro wopgrvkl qxuvpbq $c$ bxh gndns $P(X\leq c)\geq0.95$ .

$c$	$P(X\leq c)$	Meets the condition?
$5$	$0.882151\ldots$	Fn
$6$	$0.961399\ldots$	Flx

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