Notes 8
BIglyph1197OMIAL DISTRIBUTIOglyph1197
Bernoulli Trials
Bernoulli trials are trials that satisfy the following three conditions:
head2right
There are only two possible outcomes for each trial, called a “success” and “failure”
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The probability of a success is the same for all trials
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The trials are independent.
A binomial situation arises when
n
independent trials (Bernoulli trials) of an experiment are
performed.
Examples
:
checkbld
Tossing a coin 6 times where obtaining a head on a
single
toss is called the success and
getting a tail is a failure.
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Rolling a die 10 times, where getting a 6 face up on a
single
throw is a success and not
getting a 6 is a failure.
Fixed number of trials
Binomial Conditions
:
Trials are independent
Only two possible outcomes at each trial
Probability of success (p) is the same for all trials
If
X =
the
number of successes
that occur in a
fixed number of Bernoulli trials
, then the
appropriate probability model for X is the
BIglyph1197OMIAL MODEL.
Note
: X is discrete with sample space S = {0,1,2,…n}.
Therefore a Binomial Random Variable is used to describe the possible number of times
that a
particular event will occur
in a sequence of trials
that are independent and identical.
We say
X ~ Bin(n,p)
glyph817ote: n and p are the parameters used to define the Binomial distribution.
Examples
:
(i)
A new drug is administered to a group of 20 patients. The probability of improvement
is 0.7 for each patient.
Let X = the number of patients who improved out of the 20 patients.
X is a Binomial Random Variable.
_______________________
(ii)
A bag contains 2 red, 1 blue and 3 green balls. A person draws a ball with
replacement on 10 occasions.
Let G = the number of times a green ball is selected in the 10 draws.
G ________ a Binomial Random Variable.
_______________________
(iii)
If the person decides to draw without replacement, is G still a Binomial r.v.?

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Notes 8
Calculating Probabilities for a Binomial Distribution
A probability distribution usually has a distribution function associated with it that can be used to
calculate probabilities for that particular type of distribution. The Binomial distribution is no
exception.
If
X
is a Binomial random variable with
n
trials, probability of success
p
(therefore probability of
failure
q=1-p
), then finding the probability that
k
successes occur requires two things:
(1) The probability of any outcome in which there are
k
successes (and
n – k
failures):
(p× p× …×p)
×
(
q× q×…×q
) =
p
k
q
n-k
k successes
n-k
failures
(2) The number of ways that these
k
successes can be chosen from
n
trials is:
)!
(
!

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