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Binomial Distribution Tutorial

Definition:

The Binomial Distribution is one of the discrete probability distribution. It is used

when there are exactly two mutually exclusive outcomes of a trial. These outcomes areappropriately labeled Successand Failure. The Binomial Distribution is used to obtain the

probability of observing r successes in n trials, with the probability of success on a single trial

denoted by p.

Formula:

P( ! r" ! n#rpr ($%p"n%r

where,

n ! &umber of events

r ! &umber of successful events.

p ! Probability of success on a single trial. n#r! ( n' (n%r"' " r'

$%p ! Probability of failure.

Example) Toss a coin for $* times. +hat is the probability of getting exactly heads.

-tep $) ere,

&umber of trials n ! $*

&umber of success r ! since we define getting a head as success

Probability of success on any single trial p ! /.0

-tep *) To #alculate n#rformula is used.

n#r! ( n' (n%r"' " r'

! ( $*' ($*%"' " '

! ( $*' 0' " '

! ( 12//$3// $*/ " 0/1/

! ( 422$35/ 0/1/ "

! 2*

-tep 4) 6ind pr.

pr

! /.0

! /.//5$*0

-tep 1) To 6ind ($%p"n%r#alculate $%p and n%r.

$%p ! $%/.0 ! /.0

n%r ! $*% ! 0

-tep 0) 6ind ($%p"n%r.

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! /.00! /./4$*0

-tep 3) -olve P( ! r" ! n#rpr ($%p"n%r

! 2* 7 /.//5$*0 7 /./4$*0

! /.$2440240

The probability of getting exactly heads is /.$2

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The Poisson distributionis a discrete distribution. It is often used as a model for the

number of events (such as the number of telephone calls at a business, number of

customers in waiting lines, number of defects in a given surface area, airplane arrivals,

or the number of accidents at an intersection) in a specific time period. It is also usefulin ecological studies, e.g., to model the number of prairie dogs found in a square mile

of prairie. The major difference between Poisson and Binomial distributions is that the

Poisson does not have a fied number of trials. Instead, it uses the fied interval of

time or space in which the number of successes is recorded.

Parameters)The mean is . The variance is .

is the parameter which indicates the average number of events in the given time interval.

Ex.1. On an average Friday, a waitress gets no tip from !ustomers. Find t"e probability

t"at s"e will get no tip from # !ustomers t"is Friday.

The waitress averages ! customers that leave no tip on "rida#s$ $% !.

&andom 'ariable $ The number of customers that leave her no tip this "rida#.

e are interested in P( % *).

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Ex. % During a typi!al football game, a !oa!" !an expe!t &.% in'uries. Find t"e probability

t"at t"e team will "ave at most 1 in'ury in t"is game.

+ coach can epect .- injuries $ $% .-.

&andom 'ariable $ The number of injuries the team has in this game.

e are interested in .

Ex. 3. A small life insurance company has determined that on the averae it receives !

death claims per day. Find the pro"a"ility that the company receives at least seven death

claims on a randomly selected day.

P(x " %$ /P(x 03" % /.42432

Ex. #. The num"er of traffic accidents that occurs on a particular stretch of road durin a

month follo$s a Poisson distri"ution $ith a mean of %.#. Find the pro"a"ility that less

than t$o accidents $ill occur on this stretch of road durin a randomly selected month.

P(x 1*" %P(x %/" 2P(x %$" % /.///53/

Poisson distri"ution examples

$. The number of road construction pro9ects that ta:e place at any one time in a certain city

follows a Poisson distribution with a mean of 4. 6ind the probability that exactly five roadconstruction pro9ects are currently ta:ing place in this city. (/.$//5$2"

*. The number of road construction pro9ects that ta:e place at any one time in a certain city

follows a Poisson distribution with a mean of . 6ind the probability that more than four road

construction pro9ects are currently ta:ing place in the city. (/.5*//5"

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4. The number of traffic accidents that occur on a particular stretch of road during a month

follows a Poisson distribution with a mean of .3. 6ind the probability that less than three

accidents will occur next month on this stretch of road. (/./$50"

1. The number of traffic accidents that occur on a particular stretch of road during a month

follows a Poisson distribution with a mean of . 6ind the probability of observing exactly three

accidents on this stretch of road next month. (/./0*$*2"

0. The number of traffic accidents that occur on a particular stretch of road during a month

follows a Poisson distribution with a mean of 3.5. 6ind the probability that the next two months

will both result in four accidents each occurring on this stretch of road. (/.//2513"

3. -uppose the number of babies born during an 5/hour shift at a hospital;s maternity wing

follows a Poisson distribution with a mean of 3 an hour. 6ind the probability that five babies are

born during a particular $/hour period in this maternity wing. (/.$3/3*4"

. The university policy department must write, on average, five tic:ets per day to :eep

department revenues at budgeted levels. -uppose the number of tic:ets written per day follows

a Poisson distribution with a mean of 5.5 tic:ets per day. 6ind the probability that less than six

tic:ets are written on a randomly selected day from this distribution. (/.$*545"

5. The number of goals scored at -tate #ollege hoc:ey games follows a Poisson distribution

with a mean of 4 goals per game. 6ind the probability that each of four randomly selected

-tate #ollege hoc:ey games resulted in six goals being scored. (./////013"