Download Review of the Binomial Distribution

Review of the
Binomial Distribution
By Young Jun Choi
Definition
• The Binomial Distribution is the
distribution of ___?___
Definition
• The Binomial Distribution is the
distribution of COUNTS.
• It counts the number of successes in a
certain number of trials.
For example, if one wanted to
find out how many free throws a
basketball player makes in one
game, one would…
COUNT the number of
shots he or she made.
Wait…so does that mean the
distribution of the shots made is
binomial?
The answer is…NO!
Why isn’t it a binomial
distribution if it depends on
counts?
To answer that question, we
must look at the Binomial
Setting
The Binomial Setting
• Fixed number of n trials
• Independence
• Two possible outcomes: success or
failure
• Same probability of a success for each
observation
Going back to the example, in
what ways did it not satisfy the
binomial setting?
• First of all, there is no set number of n
trials. In a basketball game, one cannot
predict or set how many free-throws the
player is going to shoot.
• Second, there isn’t independence or a set
probability of a success in each shot they
take. The player can improve or get worse
with more shots taken.
What is a good example that
satisfies the binomial setting?
Although there are many examples
that satisfy the binomial setting, the
Coin Toss experiment is the example
we’re going to use.
How can you carry out this
experiment so it doesn’t go
against the binomial setting?
• You select how many times you want to
toss the coin.
• Decide which side (heads or tails) is going
to be the “success” when it lands.
• Make the coin “fair”, meaning that the
probability of landing either heads or tails
is .5.
• Independence is a given, unless one can
toss the coin in a way that one outcome is
favored over the other.
To do the experiment, you can...
• Toss the coins physically and record your
observations
• Run a simulation on a calculator or on a
computer, provided by your teacher.
There is a formula that can help
us figure out the probabilities of
getting a certain number of
successes in a certain number of
trials.
What is that formula?
(Binomial Probability)
n  k
p(x  k)   p (1 p)nk
k 
Can you explain this formula?
(Test Question)
n 
p(x  k)   p k (1  p) n k
k 
• To find the probability of k successes, you
n 

find 
k  , which is the number of
sequences containing k number of
successes. Then, you multiply by the
probability of k successes and probability
If we were to toss 10 coins,
the probability of getting 6
successes is...
10  6 4
p(x  6)   (.5) (.5)  .2051
 6 
How would we do this on the
calculator?
• You can go to the MATH key, then to PRB
and find the function nCr. This is the same
n 
as 
  . Put the value of n before nCr and
k 
the number of successes you want after
k
n k
nCr. Then you multiplypby(1 p)
.
• Or you can use the function binompdf.
Binompdf(# of trials, p, x)
What’s the difference between
binompdf and binomcdf?
• If we were looking for the probability of
getting 6 heads out of 10 tosses, then
binompdf only finds the likelihood of
getting 6 successes.
• Binomcdf adds up all the probability of
successes up to that certain number, 6 in
this case, of successes, starting from 0 to k.
What is the mean and the standard
deviation of the binomial
distribution?
  np
  np(1  p)
Is this distribution normal?
No, because it depends on counts and
counts are related to proportions.
Proportions are NEVER normal.
What happens when n gets so
large that it becomes awkward to
use the formula?
You use Normal Approximation for
binomial distributions.
When can we use Normal
Approximation?
• When np is greater than or equal to 10 and
n(1-p) is greater than or equal to 10.
• Once this requirement is met, you can treat
it like a normal distribution by using
normcdf on your calculator.
• But when you use normal approximation,
the probability you get is an approximation,
while the probability you get through the
formula is exact.
Summary
• Binomial distribution depends on counts
(never normal).
• FITS: the binomial setting
  k
• The formula: p(x  k)  np
(1 p)nk

  np
k
• Parameters   np(1  p)
• Probability obtained through formula: exact
• Probability obtained through
approximation: not exact